Controlled-Phase Quantum Logic Gate

ABSTRACT

A method and circuit QED implementation of a control-phase quantum logic gate UCP(θ)=diag[1, 1, 1, eiθ]. Two qubits Qi, two resonators Ra, Rb and a modulator. Q1 and Q2, each has a frequency ωqi and characterized by σzi. Ra is associated with Q1 and defined by a quantum non-demolition (QND) longitudinal coupling g1zσ1z(â†+â). Rb is integrated into Ra, the QND second longitudinal coupling is defined by Ra as g2zδ2z({circumflex over (b)}†+{circumflex over (b)}) or, when Rb is integrated into Ra, the QND second longitudinal coupling is defined by Ra as g2zσ2z(â†+â) The modulator periodically modulates, at a frequency ωm during a time t, the longitudinal coupling strengths g1z and g2z with respective signals of respective amplitudes {tilde over (g)}1 and {tilde over (g)}2. Selecting a defined value for each of t, g1z and g2z determines θ to specify a quantum logical operation performed by the gate. Q1 and Q2 are decoupled when either one of g1z and g2z is to set to 0.

PRIORITY STATEMENT UNDER

This non-provisional patent application claims priority based upon theprior U.S. provisional patent application entitled “PARAMETRICALLYMODULATED LONGITUDINAL COUPLING”, application No. 62/305,778 filed on2016-03-09 in the name of “SOCPRA Sciences et Genie s.e.c.” and basedupon the U.S. non-provisional patent application entitled “PERIODICALMODULATION OF LONGITUDINAL COUPLING STRENGTH FOR QUANTUM NON-DEMOLITIONQUBIT READOUT”, application Ser. No. 15/455,105 filed on 2017-03-09 inthe name of “SOCPRA Sciences et Genie s.e.c.”, which are bothincorporated herein in their entirety.

STATEMENT REGARDING U.S. FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The invention was made under a contract with an agency of the U.S.Government. The name of the U.S. Government agency and Governmentcontract number are: U.S. Army Research Laboratory, GrantW911NF-14-1-0078.

STATEMENT REGARDING PRIOR DISCLOSURES BY THE INVENTOR OR A JOINTINVENTOR

To the Applicant's knowledge, no public disclosure has been made, by theinventor or joint inventor or by another who obtained the subject matterpublicly disclosed directly or indirectly from the inventor or a jointinventor, more than one (1) year before the effective filing date of aninvention claimed herein.

TECHNICAL FIELD

The present invention relates to quantum computing and, moreparticularly, to information unit representation and/or manipulation inquantum computing.

BACKGROUND

Quantum computing is presented as the next computational revolution.Yet, before we get there, different problems need to be resolved. Forinstance, one needs to reliably store information in the form of aquantum bit (qubit), maintain the information reliably in the qubit andread the stored information reliably and repetitively (i.e.,non-destructive readout). Another of the challenges of quantum computingis related to logic treatment of more than one qubit without forcing adefined state (i.e., providing one or more logical gates from differentqubits in potentially overlapping states).

The present invention addresses at least partly the need for logictreatment of more than one qubit without forcing a defined state.

SUMMARY

This summary is provided to introduce a selection of concepts in asimplified form that are further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used as an aid in determining the scope of the claimed subjectmatter.

In accordance with a first set of embodiments, a first aspect of thepresent invention is directed to a circuit quantum electrodynamics(circuit QED) implementation of a control-phase quantum logic gateU_(CP)(θ)=diag [1,1,1,e^(iθ)]. The circuit QED implementation comprisestwo qubits Q_(i), two resonators R_(a), R_(b) and a modulator. A firstqubit Q₁ and a second qubit Q₂, each has a frequency ω_(qi) and beingcharacterized by {circumflex over (σ)}_(zi). The resonator R_(a) isassociated with the qubit Q₁ and defined by a resonator frequencyω_(ra), a resonator electromagnetic field characterized by â^(†) and â,a longitudinal coupling strength g_(1z), with the qubit Q₁, a quantumnon-demolition (QND) first longitudinal coupling g_(1z){circumflex over(σ)}_(1z)(â^(†)+â). The second resonator R_(b), when independent fromR_(a), is associated with the qubit Q₂ and is defined by a longitudinalresonator-resonator coupling g_(ab) with R_(a) and R_(b) is furtherdefined by a second resonator frequency ω_(rb), a second resonatorelectromagnetic field characterized by {circumflex over (b)}^(†) and{circumflex over (b)}, a second longitudinal coupling strength g_(2z)with the qubit Q₂, a QND second first longitudinal couplingg_(2z){circumflex over (σ)}_(2z)({circumflex over (b)}^(†)+{circumflexover (b)}). When the second optional resonator R_(b) is not independentfrom R_(a) and integrated into R_(a), then R_(a) is associated with thequbit Q₂, the longitudinal resonator-resonator coupling g_(ab)=1, thesecond resonator electromagnetic field is characterized by â^(†) and âwhere {circumflex over (b)}^(†)=â^(†) and {circumflex over (b)}=â, thesecond resonator frequency ω_(rb)=ω_(ra), the second longitudinalcoupling strength g_(2z) is between the qubit Q2 and R_(a), the QNDsecond first longitudinal coupling g_(2z){circumflex over(σ)}_(2z)({circumflex over (b)}^(†)+{circumflex over (b)}) is defined byR_(a) as g_(2z){circumflex over (σ)}_(2z)(â^(†)+â). a). The modulatorperiodically modulates, at a frequency ω_(m) during a time t, thelongitudinal coupling strengths g_(1z) and g_(2z) with respectivesignals of respective amplitudes {tilde over (g)}₁ and {tilde over(g)}₂. Selecting a defined value for each of t, g_(1z) and g_(2z)determines θ to specify a quantum logical operation performed by thecontrol-phase quantum logic gate. The qubit Q₁ and the qubit Q2 aredecoupled when either one of the defined value of g_(1z) and the definedvalue of g_(2z) is to set to 0.

The circuit QED implementation may further comprise a transmitter forselectively providing a modulator activation signal to the modulator foractivating the modulator for the duration t.

The circuit QED implementation may also further comprise a signalinjector providing a squeezed input to diminish a which-qubit-stateinformation. The squeezed input may optionally be a single-mode squeezedinput or a two-mode squeezed input. The signal injector may rely onbroadband squeezed centered at ω_(rb) and/or ω_(ra).

The qubit Q₁ and the qubit Q₂ may optionally be transmons eachcomprising two Josephson junctions with respectively substantiallyequivalent capacitive values and the modulator comprises aninductor-capacitor (LC) oscillator, the longitudinal coupling resultingfrom mutual inductance between the oscillator and the transmons, theoscillator varying a flux Φ₁ in the qubit Q₁ and a flux Φ₂ in the qubitQ₂. A 3-Wave mixing Josephson dipole element may optionally be used tocouple the qubit Q₁ and the resonator R_(a).

In accordance with the firsy set of embodiments, a second aspect of thepresent invention is directed to a method for specifying a quantumlogical operation performed by a control-phase quantum logic gateU_(CP)(θ)=diag[1, 1, 1, e^(iθ)]. The circuit QED implementationcomprises (I) two qubit Q_(i), where i=1 corresponds to a first qubit Q₁and i=2 corresponds to a second qubit Q₂, each having a frequency ω_(qi)and being characterized by {circumflex over (σ)}_(zi); (II) a firstresonator R_(a), associated with the qubit Q₁, defined by a firstresonator frequency ω_(ra), a first resonator electromagnetic fieldcharacterized by â^(†) and â, a first longitudinal coupling strengthg_(1z) with the qubit Q₁ and a quantum non-demolition (QND) firstlongitudinal coupling g_(1z){circumflex over (σ)}_(1z) (â^(†)+â); (III)a second resonator R_(b), such that, when the second resonator R_(b) isindependent from R_(a), R_(b) is associated with the qubit Q₂, alongitudinal resonator-resonator coupling g_(ab) is defined and R_(b) isfurther defined by: a second resonator frequency ω_(rb), a secondresonator electromagnetic field characterized by {circumflex over(b)}^(†) and {circumflex over (b)}, a second longitudinal couplingstrength g_(2z) with the qubit Q₂, a QND second first longitudinalcoupling g_(2z){circumflex over (σ)}_(2z)({circumflex over(b)}^(†)+{circumflex over (b)}) and (IV), when the second optionalresonator R_(b) is not independent from R_(a) and integrated into R_(a),R_(a) is associated with the qubit Q₂, the longitudinalresonator-resonator coupling g_(ab)=1, the second resonatorelectromagnetic field is characterized by â^(†) and â where {circumflexover (b)}^(†)=â^(†) and {circumflex over (b)}=â, the second resonatorfrequency ω_(rb)=ω_(ra), the second longitudinal coupling strengthg_(2z) is between the qubit Q₂ and R_(a), the QND second firstlongitudinal coupling g_(2z){circumflex over (σ)}_(2z)({circumflex over(b)}^(†)+{circumflex over (b)}) is defined by R_(a) as g_(2z){circumflexover (σ)}_(2z)(â^(†)+â). The method comprises periodically modulating,at a frequency ω_(m) during a time t, the longitudinal couplingstrengths g_(1z) and g_(2z) with respective signals of respectiveamplitudes {tilde over (g)}₁ and {tilde over (g)}₂, selecting a definedvalue for each of t, g_(1z) and g_(2z) thereby fixing θ to specify thequantum logical operation performed by the control-phase quantum logicgate and setting at least one of the defined value of g_(1z) and thedefined value of g_(2z) is to 0 to decouple the qubit Q₁ from the qubitQ₂.

Selecting the defined value for each oft, g_(1z) and g_(2z) mayoptionally comprise a selectively providing a modulator activationsignal to the modulator for activating the modulator for the duration t.

The method may also further comprise providing a squeezed input todiminish a which-qubit-state information. The squeezed input may be asingle-mode squeezed input, a two-mode squeezed input or may rely onbroadband squeezed centered at ω_(rb) and/or ω_(ra).

The qubit Q₁ and the qubit Q₂ may optionally be transmons eachcomprising two Josephson junctions with respectively substantiallyequivalent capacitive values and modulating may optionally be performedby an inductor-capacitor (LC) oscillator, the longitudinal couplingresulting from mutual inductance between the oscillator and thetransmons, the oscillator varying a flux Φ₁ in the qubit Q₁ and a fluxΦ₂ in the qubit Q₂. A3-Wave mixing Josephson dipole element mayoptionally be used to couple the qubit Q₁ and the resonator R_(a).

In accordance with a second set of embodiments, a first aspect of thepresent invention is directed to a circuit quantum electrodynamics(circuit QED) implementation of a quantum information unit (qubit)memory having a qubit frequency ω_(a) and holding a value {circumflexover (σ)}_(z). The circuit QED implementation comprises a resonator, amodulator and a homodyne detector. The resonator is defined by aresonator damping rate κ, a resonator frequency ω_(r), a resonatorelectromagnetic field characterized by â^(†) and â, a longitudinalcoupling strength g_(z), an output â_(out) and a quantum non-demolition(QND) longitudinal coupling g_(z){circumflex over (σ)}_(z)(â^(†)+â). Themodulator periodically modulates the longitudinal coupling strengthg_(z) with a signal of amplitude ĝ_(z) greater than or equal to theresonator damping rate κ and of frequency ω_(m) with ω_(m)±κ resonantwith ω_(r)±a correction factor. The correction factor is smaller than|ω₄/10| and the longitudinal coupling strength g_(z) varies over time(t) in accordance with g_(z)(t)=g _(z)+{tilde over (g)}_(z) cos (ω_(m)t)with g _(z) representing an average value of g_(z). The homodynedetector for measuring the value {circumflex over (σ)}_(z) of the qubitmemory from a reading of the output â_(out).

Optionally, the correction factor may be between 0 and |ω_(r)/100|. Thehomodyne detector may measure the value {circumflex over (σ)}_(z) of thequbit memory from a phase reading of the output â_(out). The signalamplitude {tilde over (g)}_(z) may be at least three (3) times greaterthan the resonator damping rate κ or at least ten (10) times greaterthan the resonator damping rate _(N).

The circuit QED implementation may optionally further comprise a signalinjector providing a single-mode squeezed input on the resonator suchthat noise on the phase reading from the output â_(out) is reduced whilenoise is left to augment on one or more interrelated characteristics ofthe output â_(out). The average value of g_(z), g _(z) may be 0 and thesingle-mode squeezed input may be QND.

The qubit memory may be a transmon comprising two Josephson junctionswith substantially equivalent capacitive values and the longitudinalmodulator may an inductor-capacitor (LC) oscillator with a phase drop δacross a coupling inductance placed between the two Josephson junctions.The longitudinal coupling results from mutual inductance between theoscillator and the transmon and the oscillator may vary a flux Φ_(x) inthe transmon. The transmon may have a flux sweet spot at integer valuesof a magnetic flux quantum Φ₀, Josephson energy asymmetry of thetransmon may be below 0.02 and Φ_(x) may vary by ±0.05Φ₀ aroundΦ_(x)=0.9. A 3-Wave mixing Josephson dipole element may optionally beused to couple the qubit and the resonator. The resonator may further bedetuned from the qubit frequency ω_(a) by |Δ|≥{tilde over (g)}_(z). Theoscillator inductance may, for instance, be provided by an array ofJosephson junctions or by one or more Superconducting QuantumInterference Device (SQUID).

In accordance with the second set of embodiments, a second aspect of thepresent invention is directed to a method for reading a value{circumflex over (σ)}_(z) stored in a quantum information unit (qubit)memory having a qubit frequency ω_(a), with a resonator defined by aresonator damping rate κ, a resonator frequency ω_(r), a resonatorelectromagnetic field characterized by â^(†) and â, a longitudinalcoupling strength g_(z), an output â_(out) and a quantum non-demolition(QND) longitudinal coupling g_(z){circumflex over (σ)}_(z)(â^(†)+â). Themethod comprises, at a modulator, periodically modulating thelongitudinal coupling strength g_(z) with a signal of amplitude {tildeover (g)}_(z) greater than or equal to the resonator damping rate κ andof frequency ω_(m) with ω_(m)±κ resonant with ω_(r)±a correction factor.The correction factor is smaller than |ω_(r)/10| and the longitudinalcoupling strength g_(z) varies over time (t) in accordance withg_(z)(t)=g _(z)+{tilde over (g)}_(z) cos (ω_(m)t) with g _(z)representing an average value of g_(z). The method also comprises, at ahomodyne detector, measuring the value {circumflex over (σ)}_(z) of thequbit memory from a reading of the output â_(out).

Optionally, the signal amplitude {tilde over (g)}_(z) may be at leastthree (3) times greater than the resonator damping rate κ or at leastten (10) times greater than the resonator damping rate κ.

The method may further comprise, from a signal injector, providing asingle-mode squeezed input on the resonator such that noise on the phasereading from the output â_(out) is reduced while noise is left toaugment on one or more interrelated characteristics of the outputâ_(out). The average value of g_(z), g _(z) may be set to 0 and thesingle-mode squeezed input may be QND.

Optionally, the qubit memory may be a transmon comprising two Josephsonjunctions with substantially equivalent capacitive values and thelongitudinal modulator comprises an inductor-capacitor (LC) oscillatorwith a phase drop δ across a coupling inductance placed between the twoJosephson junctions. The longitudinal coupling results from mutualinductance between the oscillator and the transmon and the oscillatorvaries a flux Φ_(x) in the transmon. The transmon may have a flux sweetspot at integer values of a magnetic flux quantum Φ₀, Josephson energyasymmetry of the transmon may be below 0.02 and Φ_(x) may vary by±0.05Φ₀ around Φ_(x)=0.

The method may also further comprise detuning the resonator from thequbit frequency ω_(a) by |Δ|≥{tilde over (g)}_(z). Optionally, theoscillator inductance may be provided by an array of Josephson junctionsor by one or more Superconducting Quantum Interference Device (SQUID).

BRIEF DESCRIPTION OF THE DRAWINGS

Further features and exemplary advantages of the present invention willbecome apparent from the following detailed description, taken inconjunction with the appended drawings, in which:

FIG. 1 is schematic representation of pointer state evolution in a phasespace from an initial oscillator vacuum state;

FIG. 2 is graph presenting pointer state separation in a phase space asa function of time;

FIG. 3 is a graphical representation of different signal-to-noise ratiosversus integration time different couplings;

FIG. 4 is a graphical representation of SNR versus integration time fordifferent couplings;

FIG. 5 is a graphical representation of the measurement time T requiredto reach a fidelity of 99.99%;

FIG. 6 is a graphical representation of resonator damping rate κ/2π toreach a fidelity of 99.99% in 50 ns;

FIG. 7 is a logical representation of a circuit QED implementation ofthe longitudinal coupling modulation;

FIG. 8 is a graphical representation of flux Φ_(x)/Φ₀ dependence of bothg_(z) and g_(x);

FIG. 9 is a graphical representation of transmon frequency versus flux;FIG. 9 is a graphical representation of transmon frequency versus flux;

FIG. 10 is a logical representation of physical characteristics of atransmon qubit coupling longitudinally to a resonator;

FIG. 11 is a close-up logical representation of physical characteristicsof the transmon qubit;

FIG. 12 is a logical representation of physical characteristics of amulti-qubit architecture;

FIGS. 13A, 13B, 13C and 13D are phase space representation of pointerstates for multiple qubits;

FIGS. 14A, 14B and 14D are schematic illustrations, in rotating frame,of the qubit-state dependent oscillator field in phase space;

FIGS. 15A, 15B and 15C are schematic illustrations of average gateinfidelity and gate time;.

FIG. 16 is a logical representation of physical characteristics oflongitudinal interaction with two qubits coupled to a common LCoscillator;

FIG. 17 is a logical representation of physical characteristics oflongitudinal interaction with two qubits is series; and

FIGS. 18A and 18B are tensor representations of a CZ gate supermatrix.

DETAILED DESCRIPTION

In one set of embodiments, a controlled-phase gate based on longitudinalcoupling is disclosed. The controlled-phase gate is obtained bysimultaneously modulating the longitudinal coupling strength of twoqubits to a common resonator or two coupled resonators. The resonatormay also be presented equivalently with different terms such as anoscillator, a cavity or qubit-cavity. In contrast to the more commontransversal qubit-cavity coupling, the magnitude of the resultingeffective qubit-qubit interaction does not rely on a small perturbativeparameter. As a result, this interaction strength can be made large,resulting in effective gate times and gate fidelities. The gate fidelitycan be exponentially improved by squeezing the resonator field and theapproach may be generalized to qubits coupled to separate resonators.The longitudinal coupling strength between a qubit and a resonator mayalso be referred to as longitudinal qubit-resonator interaction.Reference is made to the drawings throughout the following description.

In another set of embodiments, an effective quantum non-demolition (QND)qubit readout is disclosed by modulating longitudinal coupling strengthbetween a qubit and a resonator. The longitudinal coupling strengthbetween a qubit and a resonator may also be referred to as longitudinalqubit-resonator interaction. In one embodiment, the QND qubit readout isaccomplished by modulating longitudinal coupling between a resonator anda qubit at the resonator frequency. The resonator may also be presentedas an oscillator, a cavity or qubit cavity. The longitudinal couplingstrength then provides a qubit-state dependent signal from theresonator. This situation is fundamentally different from the standarddispersive case. Single-mode squeezing can optionally be exploited toincrease the signal-to-noise ratio of the qubit readout protocol. Anexemplary implementation of the qubit readout is provided in circuitquantum electrodynamics (circuit QED) and a possible multi-qubitarchitecture is also exemplified. Reference is made to the drawingsthroughout the following description.

For quantum information processing, qubit readout is expected to befast, of high-fidelity and ideally QND. In order to rapidly reuse themeasured qubit, fast reset of the measurement pointer states is alsoneeded. Combining these characteristics is essential to meet thestringent requirements of fault-tolerant quantum computation. Dispersivereadout relies on coupling the qubit to an oscillator acting as pointer.With the qubit modifying the oscillator frequency in a state-dependentfashion, driving the oscillator displaces its initial vacuum state toqubit-state dependent coherent states. Resolving these pointer states byhomodyne detection completes the qubit measurement. The dispersivereadout approach is used with superconducting qubits and quantum dots,and is studied in a wide range of systems including donor-based spinqubits and Majorana fermions. The same qubit-oscillator interaction isused to measure the oscillator state in cavity QED with Rydberg atoms.

Embodiments of the present invention provide parametric modulation oflongitudinal qubit-resonator interaction for a faster, high-fidelity andideally QND qubit readout with a reset mechanism. Embodiments of thepresent invention show that the signal-to-noise ratio (SNR) of the qubitreadout can be further improved with a single-mode squeezed input stateon the resonator. Like dispersive readout, the approach presented hereinis applicable to a wide variety of systems. Skilled people will readilyrecognize that the modulation principle presented herein could beapplied outside of the superconductive context.

A quantum information unit (qubit) memory is provided with a qubitfrequency ω_(a) characterized by {circumflex over (σ)}_(z). A resonatoris provided that is defined by a resonator damping rate κ, a resonatorfrequency ω_(r), a resonator electromagnetic field characterized byâ^(†) and â, a longitudinal coupling strength g_(z) and an outputâ_(out).

Conventional dispersive readout of {circumflex over (σ)}_(z) relies ontransversal qubit-resonator coupling defined with a HamiltonianH_(x)=g_(x)(â^(†)+â){circumflex over (σ)}_(z). Embodiments of thepresent invention rely on longitudinal coupling (or longitudinalinteraction) between the resonator and the qubit memory defined with aHamiltonian Ĥ_(z)=g_(z){circumflex over (σ)}_(z)(â^(†)+â). Despite theapparently minimal change, the choice of focusing the qubit readout onlongitudinal coupling improves qubit readout. First, longitudinalcoupling leads to an efficient separation of the pointer states. Indeed,Ĥ_(z) is the generator of displacement of the oscillator field with aqubit-state dependent direction.

FIG. 1 of the drawings provides a schematic representation of pointerstate evolution of the intra-resonator field â in a phase space from aninitial resonator vacuum state. FIG. 2 provides a corresponding graphpresenting pointer state separation in a phase space as a function oftime (t). Full lines 1000 relate to longitudinal modulation readoutwhile dispersive readout for a dispersive shift

$\chi = \frac{\kappa}{2}$

is illustrated by the dashed lines 1100. Evolution from the initialresonator vacuum state is illustrated in phase space by lines 1000 and1100 after a certain time (t). Discrete times and steady-state (t→∞) areprovided for illustrative purposes with similar times being depictedusing similar hatching patterns on lines 1000 and 1100. Circle sizearound at each time are used to illustrate uncertainty on thecorresponding measured value, but are not drawn to scale on FIG. 1. Ascan be appreciated, dispersive readout illustrated by the dashed lines1100 on FIG. 1 provides for complex path in phase space and poorseparation of the pointer states at short times.

For this reason, even for identical steady-state separation of thepointers, longitudinal readout is significantly faster than itsdispersive counterpart. On FIG. 2, pointer state separation is depictedfor the resonator output field â_(out) as a function of time t. Verticaldashed lines (101 to 104) correspond to the same lines in FIG. 1.

The results depicted on FIG. 1 may be obtained using a modulatorperiodically modulating the longitudinal coupling strength g_(z) with asignal of amplitude {tilde over (g)}_(z) greater than or equal to theresonator damping rate κ and of frequency ω_(m) with ω_(m)±κ resonantwith ω_(r)±a correction factor. The correction factor is <ω_(r)/10, andmay for instance be between 0 and ω_(r)/100. In some embodiments, {tildeover (g)}_(z) is at least 3 times κ or at least 10 times κ. Thelongitudinal coupling strength g_(z) varies over time (t) in accordancewith g_(z)(t)=g _(z)+{tilde over (g)}_(z) cos (ω_(m)t) with g _(z)representing an average value of g_(z). Using a homodyne detector, thevalue {circumflex over (σ)}_(z) of the qubit memory is measured from aphase reading of the output â_(out).

FIG. 3 shows a logic depiction of an exemplary readout—reset cycle.After a measurement time τ, longitudinal modulation amplitude {tildeover (g)}_(z) is reversed (i.e., −{tilde over (g)}_(z)) during a time τto move the pointer state to the origin irrespective of the qubit state.

As can be appreciated, larger pointer state separations is achievedusing a longitudinal modulator because Ĥ_(z) commutes with the value ofthe qubit {circumflex over (σ)}_(z) (also referred to as the measuredqubit observable), resulting in an ideally QND readout. The situation isdifferent from the dispersive case because (Ĥ_(x)|{circumflex over(σ)}_(z))≈0. In the dispersive regime, where the qubit-resonatordetuning Δ is large with respect to g_(x), non-QNDness manifests itselfwith Purcell decay and with the experimentally observedmeasurement-induced qubit transitions. For these reasons, the resonatordamping rate κ cannot be made arbitrarily large using and themeasurement photon number n is typically kept well below the criticalphoton number n_(crit)=(Δ/2g_(x))². As a result, dispersive readout istypically slow (small κ) and limited to poor pointer state separation(small n).

Under longitudinal coupling, the qubit-resonator Hamiltonian reads(ℏ/2pi) Ĥ_(z)=ω_(r)â^(†)â+½ω_(a){circumflex over (σ)}z+g_(z){circumflexover (σ)}_(z)(â^(†)+â) (referred to as Equation 1 hereinafter).

In steady-state, Equation 1 leads to a qubit-state dependentdisplacement of the resonator field amplitude

$\pm {\frac{g_{z}}{\omega_{r} + \frac{i\; \kappa}{2}}.}$

A static longitudinal interaction is therefor of no consequence for thetypical case where ω_(r)«g_(z), κ.

It is proposed herein to render the longitudinal interaction resonantduring qubit readout by modulating the longitudinal coupling at theresonator frequency: g_(z)(t)=g _(z)+{tilde over (g)}_(z) cos (ω_(m)t).From the perspective of the longitudinal coupling and neglectingfast-oscillating terms, the following Equation 2 is obtained: {tildeover (H)}_(z)=½{tilde over (g)}_(z){circumflex over (σ)}_(z)(a^(†)+â)

From Equation 2, it can be appreciated that a large qubit-statedependent displacement

$\pm \frac{{\overset{\sim}{g}}_{z}}{\kappa}$

is realized. Even with a conservative modulation amplitude {tilde over(g)}_(z)˜10κ, the steady-state displacement corresponds to 100 photonsand the two qubit states are easily distinguishable by homodynedetection. With this longitudinal coupling, there is no concept ofcritical photon number and a large photon population is therefore notexpected to perturb the qubit. Moreover, as already illustrated in FIG.1, the pointer states take an improved path in phase space towards theirsteady-state separation. As shown in FIG. 2, this leads to a largepointer state separation at short times.

The consequences of using longitudinal coupling for qubit measurementcan be quantified with the signal-to-noise ratio (SNR). The SNR quantityis evaluated using {circumflex over (M)}(τ)=√κ∫₀ ^(τ)δt[â^(†)_(out)(t)+â_(out)(t)] the measurement operator for homodyne detection ofthe output signal â_(out) with a measurement time τ. The signal isdefined as |

{circumflex over (M)}

₁−

{circumflex over (M)}

₀| where {0,1} refers to qubit state, while the imprecision noise is [

{circumflex over (M)}_(N1) ²(τ)

°

{circumflex over (M)}_(N0) ²(τ)

]^(1/2) with {circumflex over (M)}_(N)={circumflex over (M)}−

{circumflex over (M)}

.

Combining these expressions, the SNR for the longitudinal case reads inaccordance with Equation 3:

${SNR}_{z} = {\sqrt{8}\frac{{\overset{\sim}{g}}_{z}}{\kappa}{\sqrt{\kappa \; \tau}\left\lbrack {1 - {\frac{2}{\kappa \; \tau}\left( {1 - e^{{- \frac{1}{2}}\kappa \; \tau}} \right)}} \right\rbrack}}$

This is to be contrasted to SNR_(χ) for dispersive readout with driveamplitude ϵ and optimal dispersive coupling χ=g_(x) ²/Δ=κ/2 inaccordance with Equation 4:

${SNR}_{\chi} = {\sqrt{8}\frac{ \in }{\kappa}{\sqrt{\kappa\tau}\left\lbrack {1 - {\frac{2}{\kappa \; \tau}\left( {1 - {e^{{- \frac{1}{2}}\kappa \; \tau}\cos \frac{1}{2}\kappa \; \tau}} \right)}} \right\rbrack}}$

Both expressions have a similar structure, making very clear the similarrole of {tilde over (g)}_(z) and E, except for the cosine in Equation(4) that is a signature of the complex dispersive path in phase space.For short measurement times κτ«1, a favorable scaling is obtained forlongitudinal modulation readout with SNR_(z)∝SNR_(χ)/κτ.

FIG. 4 shows a graphical representation of SNR versus integration timefor longitudinal (107) and dispersive without Purcell decay (105)coupling. Even for equal steady-state separation ({tilde over(g)}_(z)=ϵ), shorter measurement time are obtained for longitudinalcoupling. SNR is shown in units of {tilde over (g)}_(z)/κ as a functionof integration time τ. Longitudinal coupling (107) is compared todispersive coupling (105) with χ=κ/2 for the same steady-stateseparation, |{tilde over (g)}_(z)|=|ϵ|. Line (106) accounts for Purcelldecay in dispersive readout. Line (108) shows exponential improvementobtained for when a single-mode squeezed input state with e²⁴=100 (20dB).

FIG. 5 provides a graphical representation of the measurement time τrequired to reach a fidelity of 99.99% as a function of {tilde over(g)}_(z)/κ (or ϵ/κ for the dispersive case). When taking into accountthe non-perturbative effects that affect the QNDness of dispersivereadout, the potential of the present approach is made even clearer.Line (106) of FIG. 4 and FIG. 5 correspond to the dispersive case withPurcell decay. In this more realistic case, longitudinal readoutoutperforms its counterpart at all times. FIG. 6 provides a graphicalrepresentation of resonator damping rate κ/2π to reach a fidelity of99.99% in τ=50 ns versus intra-resonator photon number n=({tilde over(g)}_(z)/κ)²=(ϵ/κ)². Squeezing (108) helps in further reducing therequired photon number or resonator damping rate. The squeeze strengthis optimized for each κ, with a maximum set to 20 dB reached close toκ/2π=1 MHz In FIGS. 5 and 6, results for the dispersive readout arestopped at the critical photon number obtained for a drive strengthϵ_(crit)=Δ/√8g_(x) for g_(x)/Δ= 1/10.

Up to this point, equal pointer state separation has been assumed forthe longitudinal and the dispersive readouts. As already mentioned,dispersive readout is, however, limited to measurement photon numberswell below n_(crit). This is taken into account in FIGS. 5 and 6 bystopping the dispersive curves at n_(crit) (black circle) assuming thetypical value g_(x)/Δ= 1/10. FIG. 5 illustrates that only longitudinalreadout allows for measurement times <1/κ. This is moreover achieved forreasonable modulation amplitudes with respect to the cavity linewidth Asillustrated, the longitudinal coupling strength g_(z) with having asignal of amplitude {tilde over (g)}_(z) at least three (3) timesgreater than the resonator damping rate κ still allows for qubitreadout. On FIG. 6, the resonator damping rate vs photon number requiredto reach a fidelity of 99.99% in υ=50 ns is illustrated. Note that line(106) corresponding to dispersive with Purcell is absent from this plot.With dispersive readout, it appears impossible to achieve the abovetarget fidelity and measurement time in the very wide range ofparameters of FIG. 6. On the other hand, longitudinal readout with quitemoderate values of κ and n provide meaningful results. Further speedupsare expected with pulse shaping and machine learning. Because thepointer state separation is significantly improved even at short time,the latter approach should be particularly efficient.

To allow for rapid reuse of the qubit, the resonator should be returnedto its grounds state ideally in a time «1/κ after readout. A pulsesequence achieving this for dispersive readout has been proposed but isimperfect because of qubit-induced nonlinearity deriving from Ĥ_(x).

As illustrated in FIG. 3, with an approach based on longitudinalmodulation as proposed herein, resonator reset is realized by invertingthe phase of the modulation. Since Ĥ_(z) does not lead to qubit-inducednonlinearity, the reset remains ideal. In practice, reset can also beshorter than the integration time. It is also interesting to point outthat longitudinal modulation readout saturates the inequalityΓ_(φm)≥Γ_(meas) linking the measurement-induced dephasing rate Γ_(φm) tothe measurement rate Γ_(meas) and is therefore quantum limited.

Another optional feature of the longitudinal modulation readout toimprove SNR (theoretically exponentially) by providing a single-modesqueezed input state on the resonator. The squeeze axis is chosen to beorthogonal to the qubit-state dependent displacement generated byg_(z)(t). referring back to the example of FIG. 1, squeezing wouldprovide for a more defined value along the vertical axis. Since thesqueeze angle is unchanged under evolution with Ĥ_(z), the imprecisionnoise is exponentially reduced along the vertical axis while theimprecision noise is left to augment along the horizontal axis. Thesignal-to-noise ratio becomes e^(r)SNR_(z), with r the squeezeparameter. This exponential enhancement is apparent from line (108)depicted in FIG. 4 and in the corresponding reduction of the measurementtime in FIG. 5. Note that by taking g _(z)=0, the resonator field can besqueezed prior to measurement without negatively affecting the qubit.

The exponential improvement is in contrast to standard dispersivereadout where single-mode squeezing can lead to an increase of themeasurement time. Indeed, under dispersive coupling, the squeeze angleundergoes a qubit-state dependent rotation. As a result, both thesqueezed and the anti-squeezed quadrature contributes to the imprecisionnoise. It is to be noted that the situation can be different in thepresence of two-mode squeezing where an exponential increase in SNR canbe recovered by engineering the dispersive coupling of the qubit to twocavities.

While the longitudinal modulation approach is very general, a circuitQED implementation 700 is discussed in greater details hereinafter withreference to FIG. 7. Longitudinal coupling of a flux or transmon qubitto a resonator of the LC oscillator type may result from the mutualinductance between a flux-tunable qubit and the resonator. Anotherexample focuses on a transmon qubit phase-biased by the oscillator. FIG.7 shows a schematically lumped version of an exemplary circuit QEDimplementation 700 considering the teachings of the present disclosure.In practice, the inductor L can be replaced by a Josephson junctionarray, both to increase the coupling and to reduce the qubit'sflux-biased loop size. Alternatives (e.g., based on a transmission-lineresonator) may also be realized as it is explored with respect to otherembodiments described hereinbelow.

The Hamiltonian of the circuit of FIG. 7 is similar to that of aflux-tunable transmon, but where the external flux Φ_(x) is replaced byΦ_(x)+δ with δ the phase drop at the oscillator. Taking the junctioncapacitances Cs to be equal and assuming for simplicity that Z₀/R_(K)«1,with Z₀=√L/C and R_(K) the resistance quantum, the Hamiltonian of thecircuit QED 700 may be expressed as Ĥ=Ĥ_(r)+Ĥ_(q)+Ĥ_(qr), withH_(r)=ω_(r)â^(†)â resenting the oscillator Hamiltonian andĤ_(q)=½ω_(a){circumflex over (σ)}_(z) presenting the Hamiltonian of aflux transmon written here in its two-level approximation.

The Hamiltonian of the qubit-oscillator interaction (or longitudinalcoupling strength) takes the form Ĥ_(qr)=g_(x){circumflex over(σ)}_(z)(â^(†)+â)+g_(z){circumflex over (σ)}_(z)(â^(†)+â) when Equation5 and Equation 6 are satisfied:

${g_{z} = {{- \frac{E_{J}}{2}}\left( \frac{2E_{C}}{E_{J}} \right)^{1/2}\sqrt{\frac{\pi \; Z_{0}}{R_{K}}}{\sin \left( \frac{{\pi\Phi}_{x}}{\Phi_{0}} \right)}}},{g_{x} = {{{dE}_{J}\left( \frac{2E_{C}}{E_{J}} \right)}^{1/4}\sqrt{\frac{\pi \; Z_{0}}{R_{K}}}{\cos \left( \frac{{\pi\Phi}_{x}}{\Phi_{0}} \right)}}}$

where E_(J)is the mean Josephson energy, d the Josephson energyasymmetry and E_(C) the qubit's charging energy. Skilled person willreadily be able to locate expressions for these quantities in terms ofthe elementary circuit parameters. In the circuit QED implementation700, E_(J1)=E_(J)(1+d)/2 and E_(J2)=E_(J)(1−d)/2 with dϵ[0,1]. Aspurposely pursued, the transverse coupling g_(x) vanishes exactly ford=0, leaving only longitudinal coupling g_(z). Because longitudinalcoupling is related to the phase bias rather than inductive coupling,g_(z) can be made large.

For example, with the realistic values E_(J)/h=20 GHz, E_(J)/E_(C)=67and Z₀=50Ω, g_(z)/2π≈r=135 MHz×sin (πΦ_(x)/Φ₀) where Φ₀ represents themagnetic flux quantum. FIG. 8 shows a graphical representation of fluxΦ_(z)/Φ₀ dependence of both g_(z) (full line) and g_(x) with d=0(dash-dotted line) and d=0.02 (dashed line). Modulating the flux by0.05Φ₀ around Φ_(x)=0, it follows that g _(z)=0 and {tilde over(g)}_(z)/2π˜21 MHz Conversely, only a small change of the qubitfrequency of ˜40 MHz is affected, as can be appreciated from FIG. 9showing transmon frequency versus flux in accordance with the precedingexemplary parameters. Importantly, this does not affect the

SNR.

As can be appreciated, a finite g_(x) for d≠0. On FIG. 8, a realisticvalue of d=0.02 and the above parameters, g_(x)/2π≈13 MHz×cos(πΦ_(x)/Φ₀). The effect of this unwanted coupling can be mitigated byworking at large qubit-resonator detuning Δ where the resultingdispersive interaction χ=g_(x) ²/2Δ can be made very small. For example,the above numbers correspond to a detuning of Δ/2π=3 GHz where χ/2π˜5.6kHz. It is important to emphasize that, contrary to dispersive readout,the longitudinal modulation approach is not negatively affected by alarge detuning.

When considering higher-order terms in Z₀/R_(K), the Hamiltonian of thecircuit QED 700 exemplified in FIG. 7 contains a dispersive-likeinteraction χ_(z)â^(†)â{circumflex over (σ)}_(z) even at d=0. For theparameters already used above, χ_(z)/2π˜5.3 MHz, a value that is notmade smaller by detuning the qubit from the resonator. However since itis not derived from a transverse coupling, χ_(z) is not linked to anyPurcell decay. Moreover, it does not affect SNR_(z) at small measurementtimes.

In the absence of measurement, g _(z)=ĝ_(z)=0 and the qubit mayadvantageously be parked at its flux sweet spot (e.g., integer values ofΦ₀). Dephasing due to photon shot noise or to low-frequency flux noiseis therefore expected to be minimal. Because of the longitudinalcoupling, another potential source of dephasing is flux noise at theresonator frequency which will mimic qubit measurement. However, giventhat the spectral density of flux noise is proportional to 1/f even athigh frequency, this contribution is negligible.

In the circuit QED implementation, the longitudinal readout can also berealized with a coherent voltage drive of amplitude ϵ(t) applieddirectly on the resonator, in place of a flux modulation on the qubit.Taking into account higher-order terms in the qubit-resonatorinteraction, the full circuit Hamiltonian without flux modulation can beapproximated to (g_(z)=0)

Ĥ=ω _(r) â ^(†â+)½ω_(a){circumflex over (σ)}_(z)+χ_(z) â ^(†)â{circumflex over (σ)} _(z) +iγ(t)(â ^(†) −â)

the well-known driven dispersive Hamiltonian where the AC-Stark shiftinteraction originating from the higher-order longitudinal interactionis given by

$\chi_{z} = {{- \frac{\sqrt{E_{J}E_{C}}}{h}}\frac{\pi \; Z_{r}}{R_{K}}}$

Assuming a drive resonant with the resonator frequency ω_(r) with phaseφ=0 for simplicity, in the rotating frame the system Hamiltonian becomes(neglecting fast-rotating terms)

$\hat{H} + {\chi_{z}{\hat{a}}^{\dagger}\hat{a}\; {\hat{\sigma}}_{z}} + {i\frac{\epsilon}{2}\left( {{\hat{a}}^{\dagger} - \hat{a}} \right)}$

Under a displacement transformation D(α)âD^(†)(α)→â−α and including theresonator dissipation, the following is obtained:

$\hat{H} + {{\chi \left( {{\hat{a}}^{\dagger} - \alpha^{*}} \right)}\left( {\hat{a} - \alpha} \right){\hat{\sigma}}_{z}} + {i\frac{\epsilon}{2}\left( {{\hat{a}}^{\dagger} - \hat{a}} \right)} + {i\frac{\kappa}{2}\left( {{\alpha \; {\hat{a}}^{\dagger}} - {\alpha^{*}\hat{a}}} \right)}$

Finally, choosing α=ϵ/κ, the system is now simplifed to

Ĥ=χ _(z) â ^(†)â{circumflex over (σ)}_(z) +g′ _(z)(â ^(†) +â){circumflexover (σ)}_(z)

with an effective (driven) longitudinal interaction with strengthg′_(z)=χ_(z)ϵ/κ. In a regime of large voltage drive amplitudes withϵ/κ»1, the voltage drive performs the ideal longitudinal readout as theresidual dispersive effects are mitigated g′_(z)»χ_(z). As mentionedearlier, the absence of Purcell decay and of any critical number ofphotons in the system allows to push the standard dispersive readoutmechanism towards the ideal limit of the longitudinal readout.

A possible multi-qubit architecture consists of qubits longitudinallycoupled to a readout resonator (of annihilation operator â_(z)) andtransversally coupled to a high-Q bus resonator (â_(x)). The Hamiltoniandescribing this system is provided by Equation 7:

$\hat{H} = {{\omega_{rz}{\hat{a}}_{z}^{\dagger}{\hat{a}}_{z}} + {\omega_{rz}{\hat{a}}_{x}^{\dagger}{\hat{a}}_{x}} + {\sum\limits_{j}{\frac{1}{2}\omega_{aj}{\hat{\sigma}}_{zj}}} + {\sum\limits_{j}{g_{zj}{{\hat{\sigma}}_{zj}\left( {{\hat{a}}_{z}^{\dagger} + {\hat{a}}_{z}} \right)}}} + {\sum\limits_{j}{g_{xj}{{\hat{\sigma}}_{xj}\left( {{\hat{a}}_{x}^{\dagger} + {\hat{a}}_{x}} \right)}}}}$

Readout can be realized using longitudinal coupling while logicaloperations via the bus resonator. Alternative architectures, e.g.,taking advantage of longitudinal coupling may also be proposed. Here forinstance, taking g_(zj)(t)=g _(z)+{tilde over (g)}_(z) cos(ω_(r)t+φ_(j)), the longitudinal coupling, from the perspective of thelongitudinal interaction and neglecting fast-oscillating terms, isrepresented by Equation 8: Ĥ_(z)=(½g _(z)Σ_(j){circumflex over(σ)}_(zj)e^(−iφj))â_(z)+H.c.

This effective resonator drive displaces the field to multi-qubit-statedependent coherent states. For two qubits and taking φ_(j)=jπ/2 leads tothe four pointer states separated by 90° from each other or, in otherwords, to an optimal separation even at short times. Other choices ofphase lead to overlapping pointer states corresponding to differentmulti-qubit states. Examples are φ_(j)=0 for which |01¢ and |10

are indistinguishable, and φ_(j)=jπ where |00

and |11

are indistinguishable. However, these properties may be exploited tocreate entanglement by measurement. As another example, with 3 qubitsthe GHZ state may be obtained with φ_(j)=j2π/3.

In the following pages, additional sets of embodiments are presented. Ina first additional set of exemplary embodiments, longitudinal couplingis considered (A). In a second additional set of exemplary embodiments,longitudinal coupling with single-mode squeezed states is presented (B).Standard dispersive coupling (C) as well as innovative longitudinalcoupling in presence of transverse coupling in the dispersive regime (D)are also considered.

A. Longitudinal coupling

1. Modulation at the Resonator Frequency

The first set of embodiments (A) considers a qubit longitudinallycoupled to a resonator with the Hamiltonian:

Ĥ=ω _(r) â ^(†) â+½ω{circumflex over (σ)}_(z) +[g _(z)(t)â ^(†) +g_(z)*(t)â]{circumflex over (σ)} _(z).   (S1)

In this expression, ω_(r) is the resonator frequency, ω_(a) the qubitfrequency and g_(z) is the longitudinal coupling that is modulated atthe resonator frequency:

g _(z)(t)= g _(z) +|{tilde over (g)} _(z)| cos (ω_(r) t+φ).   (S2)

In the interaction picture and using the Rotating-Wave Approximation(RWA), the above Hamiltonian simplifies to

Ĥ=½[{tilde over (g)} _(z) â ^(†) +{tilde over (g)} _(z) *â]{circumflexover (σ)} _(z),   (S3)

where {tilde over (g)}_(z)≡|{tilde over (g)}_(z)|e^(iφ) is themodulation amplitude. From Equation (S3), it is clear that the modulatedlongitudinal coupling plays the role of a qubit-state dependent drive.The Langevin equation of the cavity field simply reads

{dot over (â)}=−i½{tilde over (g)} _(z){circumflex over (σ)}_(z)−½κâ−√κâ_(in),   (S4)

where â_(in) is the input field. Taking this input to be the vacuum, theinput correlations are then defined by:

Using the input-output boundary â_(out)=â_(in)+√κâ, it integration ofthe Langevin equation leads to

{circumflex over (â)}_(in)(t)â_(in) ^(†)(t′)

=[â_(in)(t), â_(in) ^(†)(t′)]=δ(t−t′)

$\begin{matrix}{{{{\alpha_{out}(t)} = {{- \frac{i\; {\overset{\sim}{g}}_{z}}{\sqrt{\kappa}}}{{\langle{\hat{\sigma}}_{z}\rangle}\left\lbrack {1 - e^{{- \frac{1}{2}}\kappa \; t}} \right\rbrack}}},{{{\hat{d}}_{out}(t)} = {{{\hat{d}}_{in}(t)} - {\kappa {\int_{- \infty}^{t}{{dt}^{\prime}e^{{- \frac{1}{2}}{\kappa {({t - t^{\prime}})}}}{{\hat{d}}_{in}\left( t^{\prime} \right)}}}}}},}\ } & \left( {S\; 5} \right)\end{matrix}$

where α_(out)=

â_(out)

stands for the output field mean value and {circumflex over(d)}_(out)=â_(out)−α_(out) its fluctuations. Because here thequbit-dependent drive comes from modulations of the coupling, and notfrom an external coherent drive, there is no interference between theoutgoing and the input fields. As a result, α=α_(out)/√κ and theintracavity photon number evolves as

$\begin{matrix}{{\langle{{\hat{a}}^{\dagger}\hat{a}}\rangle} = {{\frac{{{\overset{\sim}{g}}_{z}}^{2}}{\kappa^{2}}\left\lbrack {1 - e^{{- \frac{1}{2}}\kappa \; t}} \right\rbrack}^{2}.}} & \left( {S\; 6} \right)\end{matrix}$

The measurement operator corresponding to homodyne detection of theoutput signal with an integration time τ and homodyne angle φ_(h) is

{circumflex over (M)}(τ)=√κ∫₀ ^(τ) dt[â _(out) ^(†)(t)e ^(iφ) ^(h) +â_(out)(t)e ^(−iφ) ^(h) ].   (S7)

The signal for such a measurement is

M

while the noise operator is {circumflex over (M)}_(N)={circumflex over(M)}−

{circumflex over (M)}

In the presence of a qubit, the measurement signal is then

$\begin{matrix}{{{\langle\hat{M}\rangle}_{1} - {\langle\hat{M}\rangle}_{0}} = {4{{\overset{\sim}{g}}_{z}}{\sin \left( {\phi - \varphi_{h}} \right)}{{\tau \left\lbrack {1 - {\frac{2}{\kappa \; \tau}\left( {1 - e^{{- \frac{1}{2}}{\kappa\tau}}} \right)}} \right\rbrack}.}}} & ({S8})\end{matrix}$

On the other hand, the measurement noise is equal to

{circumflex over (M)}_(N) ²(τ)

=κτ. Combining these two expressions, the signal-to-noise ratio (SNR)then reads

$\begin{matrix}\begin{matrix}{{SNR}^{2} \equiv \frac{{{{\langle\hat{M}\rangle}_{1} - {\langle\hat{M}\rangle}_{0}}}^{2}}{{\langle{{\hat{M}}_{N\; 1}^{2}(\tau)}\rangle} + {\langle{{\hat{M}}_{N\; 0}^{2}(\tau)}\rangle}}} \\{= {\frac{8{{\overset{\sim}{g}}_{z}}^{2}}{\kappa^{2}}{\sin^{2}\left( {\phi - \varphi_{h}} \right)}\kappa \; {{\tau \left\lbrack {1 - {\frac{2}{\kappa \; \tau}\left( {1 - e^{{- \frac{1}{2}}{\kappa\tau}}} \right)}} \right\rbrack}^{2}.}}}\end{matrix} & \begin{matrix}({S9}) \\\begin{matrix}\; \\({S10})\end{matrix}\end{matrix}\end{matrix}$

The SNR is optimized by choosing the modulation phase φ and the homodyneangle such that φ−φh=modπ.With this choice, the optimized SNR finallyreads:

$\begin{matrix}{{SNR} = {\sqrt{8}\frac{{\overset{\sim}{g}}_{z}}{\kappa}{{\sqrt{\kappa\tau}\left\lbrack {1 - {\frac{2}{\kappa \; \tau}\left( {1 - e^{{- \frac{1}{2}}\kappa \; \tau}} \right)}} \right\rbrack}.}}} & \left( {S\; 11} \right)\end{matrix}$

At long measurement times (τ»1/κ), the signal-to-noise ratio evolves as

${SNR} = {\sqrt{8}\frac{{\overset{\sim}{g}}_{z}}{\kappa}\sqrt{\kappa\tau}}$

while in the more experimentally interesting case of short measurementtimes leads to

${SNR} =^{\frac{1}{\sqrt{2}}\frac{{\overset{\sim}{g}}_{z}}{\kappa}{({\kappa \; \tau})}^{3/2}}.$

In short, the SNR increases as τ^(3/2), much faster than in thedispersive regime where the SNR rather increases as τ^(5/2), as willbecome apparent in (C) below.

2. Measurement and Dephasing Rates

To evaluate the measurement-induced dephasing rate, a polaron-typetransformation is applied on Hamiltonian from (S3) consisting of adisplacement of a â by −i{tilde over (g)}_(z){circumflex over(σ)}_(z)/κ. this transformation, the cavity decay Lindbladian κ

[â]{circumflex over (ρ)}=

[â]ρ=âρâ^(†)−½{â^(†)â, ρ}, leads to ½Γ_(φm)

[{circumflex over (σ)}_(z)]{circumflex over (ρ)} whereΓ_(φm)=2[ĝ_(z)]²/κ is the measurement-induced dephasing. On the otherhand, the measurement rate is obtained from the SNR asΓ_(meas)=SNR²/(4τ)=2[{tilde over (g)}_(z)]²/κ.

The relation between the dephasing and the measurement rate is thenΓ_(meas)=Γ_(φm). This is the bound reached for a quantum limitedmeasurement.

3. Modulation Bandwidth

A situation where the longitudinal coupling is modulated at a frequencyω_(m)≠ω_(r) is now considered. That is, g_(z)(t)=g _(z)+|{tilde over(g)}_(z)| cos (ω_(m)t+φ). Assuming that the detuning Δm=ω_(m)−ω_(r) issmall with respect to the modulation amplitude {tilde over (g)}_(z), theHamiltonian in a frame rotating at the modulation frequency now readsunder the RWA as:

Ĥ=−Δ _(m) â ^(†) â+½[{tilde over (g)} _(z) â ^(†) +{tilde over (g)} _(z)â]{circumflex over (σ)} _(a),   (S12)

The corresponding Langevin equation is then

{dot over (â)}=−i1/2{tilde over (g)} _(z){circumflex over (σ)}_(a) −iΔ_(m) â−1/2κâ−√κâ _(in),   (S13)

yielding for the output field,

$\begin{matrix}{{{\alpha_{out}(t)} = {{- \frac{i\; {\overset{\sim}{g}}_{z}\sqrt{\kappa}}{\kappa - {2i\; \Delta_{m}}}}{{\langle{\hat{\sigma}}_{z}\rangle}\left\lbrack {1 - e^{{({{i\; \Delta_{m}} - {\frac{1}{2}\kappa}})}\tau}} \right\rbrack}}},{{{\hat{d}}_{out}(t)} = {{{\hat{d}}_{in}(t)} - {\kappa {\int_{- \infty}^{t}{{dt}^{\prime}e^{{({{i\; \Delta_{m}} - {\frac{1}{2}\kappa}})}{({t - t^{\prime}})}}{{{\hat{d}}_{in}\left( t^{\prime} \right)}.}}}}}}} & \left( {S\; 14} \right)\end{matrix}$

From these expressions, the measurement signal is then

$\begin{matrix}{{{\langle\hat{M}\rangle}_{1} - {\langle\hat{M}\rangle}_{0}} = {{\frac{4{{\overset{\sim}{g}}_{z}}}{\sqrt{1 + \left( \frac{2\Delta_{m}}{\kappa} \right)^{2}}}{\sin \left\lbrack {\phi - \varphi_{h} + {\arctan \left( {2{\Delta_{m}/\kappa}} \right)}} \right\rbrack}\tau} - {\frac{8{{{\overset{\sim}{g}}_{z}}/\kappa}}{1 + \left( \frac{2\; \Delta_{m}}{\kappa} \right)^{2}}{\sin \left\lbrack {\phi - \varphi_{h} + {2{\arctan \left( {2{\Delta_{m}/\kappa}} \right)}}} \right\rbrack}} + {\frac{8{{{\overset{\sim}{g}}_{z}}/\kappa}}{1 + \left( \frac{2\; \Delta_{m}}{\kappa} \right)^{2}}{\sin \left\lbrack {\phi - \varphi_{h} + {2{\arctan \left( {2{\Delta_{m}/\kappa}} \right)}} + {\Delta_{m}\tau}} \right\rbrack}{e^{{- \frac{1}{2}}\kappa \; \tau}.}}}} & ({S15})\end{matrix}$

While the signal is changed, the noise is however not modified by thedetuning. From the above expression, given a detuning Am and ameasurement time T, there is an optimal angle φ that maximizes the SNR.

B. Longitudinal coupling with squeezing

In the second set of exemplary embodiments (B), a situation where themodulation detuning is zero and where the input field is in asingle-mode squeezed vacuum is now considered. This leaves the signalunchanged, but as will be appreciated, leads to an exponential increaseof the SNR with the squeeze parameter r. Indeed, in the frame of theresonator, the correlations of the bath fluctuations are now

$\begin{matrix}{\begin{pmatrix}{\langle{{{\hat{d}}_{in}^{\dagger}(t)}{{\hat{d}}_{in}\left( t^{\prime} \right)}}\rangle} & {\langle{{{\hat{d}}_{in}(t)}{{\hat{d}}_{in}\left( t^{\prime} \right)}}\rangle} \\{\langle{{{\hat{d}}_{in}^{\dagger}(t)}{{\hat{d}}_{in}^{\dagger}\left( t^{\prime} \right)}}\rangle} & {\langle{{{\hat{d}}_{in}(t)}{{\hat{d}}_{in}^{\dagger}\left( t^{\prime} \right)}}\rangle}\end{pmatrix} = {\begin{pmatrix}{\sinh^{2}r} & {\frac{1}{2}\sinh \; 2\; r\; e^{2i\; \theta}} \\{\frac{1}{2}\sinh \; 2\; r\; e^{{- 2}i\; \theta}} & {\cosh^{2}r}\end{pmatrix}{{\delta \left( {t - t^{\prime}} \right)}.}}} & ({S16})\end{matrix}$

where is has been assumed broadband squeezing with a squeeze angle θ.The measurement noise is then

$\begin{matrix}{{\langle{{\hat{M}}_{N}^{2}(\tau)}\rangle} = {\kappa {\int_{0}^{\tau}{{dt}{\int_{0}^{\tau}{{{dt}^{\prime}\ \left\lbrack {{\langle{{{\hat{d}}_{out}^{\dagger}(t)}{{\hat{d}}_{out}\left( t^{\prime} \right)}}\rangle} + {\langle{{{\hat{d}}_{out}(t)}{{\hat{d}}_{out}^{\dagger}\left( t^{\prime} \right)}}\rangle} + {{\langle{{{\hat{d}}_{out}(t)}{{\hat{d}}_{out}\left( t^{\prime} \right)}}\rangle}e^{{- 2}i\; \varphi_{h}}} + {{\langle{{{\hat{d}}_{out}^{\dagger}(t)}{{\hat{d}}_{out}^{\dagger}\left( t^{\prime} \right)}}\rangle}e^{2{iO}_{h}}}} \right\rbrack}.}}}}}} & ({S17})\end{matrix}$

The output-field correlations are easily obtained from

$\begin{matrix}{\begin{pmatrix}{\langle{{{\hat{d}}_{out}^{\dagger}(t)}{{\hat{d}}_{out}\left( t^{\prime} \right)}}\rangle} & {\langle{{{\hat{d}}_{out}(t)}{{\hat{d}}_{out}\left( t^{\prime} \right)}}\rangle} \\{\langle{{{\hat{d}}_{out}^{\dagger}(t)}{{\hat{d}}_{out}^{\dagger}\left( t^{\prime} \right)}}\rangle} & {\langle{{{\hat{d}}_{out}(t)}{{\hat{d}}_{out}^{\dagger}\left( t^{\prime} \right)}}\rangle}\end{pmatrix} = \begin{pmatrix}{\langle{{{\hat{d}}_{in}^{\dagger}(t)}{{\hat{d}}_{in}\left( t^{\prime} \right)}}\rangle} & {\langle{{{\hat{d}}_{in}(t)}{{\hat{d}}_{in}\left( t^{\prime} \right)}}\rangle} \\{\langle{{{\hat{d}}_{in}^{\dagger}(t)}{{\hat{d}}_{in}^{\dagger}\left( t^{\prime} \right)}}\rangle} & {\langle{{{\hat{d}}_{in}(t)}{{\hat{d}}_{in}^{\dagger}\left( t^{\prime} \right)}}\rangle}\end{pmatrix}} & \left( {S\; 18} \right)\end{matrix}$

which holds here since the drive is ‘internal’ to the cavity. As aresult))

{circumflex over (M)} _(N) ²(τ)

={cos h(2r)−sin h(2r) cos [2(φ_(h)−θ)]}κτ.   (S19)

The noise is minimized by choosing θ according to

${\theta - {\varphi \; h}} = {\frac{\pi}{2}{mod}\; {\pi.}}$

With this choice, the SNR reads

SNR(r)=e ^(r)SNR(r=0).   (S20)

The SNR is thus exponentially enhanced, leading to Heisenberg-limitedscaling.

A source of broadband Γ pure squeezing is assumed to be available. Theeffect of a field squeezing bandwidth r was already studied elsewhere,it only leads to a small reduction of the SNR for r»κ. On the otherhand, deviation from unity of the squeezing purity P leads to areduction of the SNR by 1/√P. The SNR being decoupled from theanti-squeezed quadrature, the purity simply renormalizes the squeezeparameter.

C. Dispersive coupling

For completeness, the SNR for dispersive readout is also provided, eventhough corresponding result may be found in the literature. In thedispersive regime, the qubit-cavity Hamiltonian reads

Ĥ=Φ _(r) â ^(†) â+½ω_(a){circumflex over (σ)}_(z) +χâ ^(†) â{circumflexover (σ)} _(z),   (S21)

where χ=g_(x) ²/Δ is the dispersive shift. The Langevin equation of thecavity field in the interaction picture then reads

{dot over (â)}=−iχ{circumflex over (σ)} _(z) â−½κâ−√κâ _(in).   (S22)

With a drive of amplitude ϵ=|ϵ|e^(−iφd) on the cavity at resonance, theinput field is defined by its mean a_(in)=

â_(in)

=−ϵ/√κ and fluctuations {circumflex over (d)}_(in)=â_(in)−α_(in).Integrating the Langevin equation yields

$\begin{matrix}{{{\alpha_{out}(t)} = {\frac{\epsilon}{\sqrt{\kappa}}{e^{{- i}\; \phi_{qb}{\langle{\hat{\sigma}}_{z}\rangle}}\left\lbrack {1 - {2{\cos \left( {\frac{1}{2}\phi_{qb}} \right)}e^{{{- {({{i\; \chi {\langle{\hat{\sigma}}_{z}\rangle}} + {\frac{1}{2}\kappa}})}}t} + {\frac{1}{2}i\; \phi_{gb}{\langle{\hat{\sigma}}_{z}\rangle}}}}} \right\rbrack}}},} & \left( {S\; 23} \right) \\{\mspace{79mu} {{{\hat{d}}_{out}(t)} = {{{\hat{d}}_{in}(t)} - {\kappa {\int_{- \infty}^{t}{{dt}^{\prime}e^{{- {({{i\; \chi {\langle{\hat{\sigma}}_{z}\rangle}} - {\frac{1}{2}\kappa}})}}{({t - t^{\prime}})}}{{{\hat{d}}_{in}\left( t^{\prime} \right)}.}}}}}}\ } & \left( {S\; 24} \right)\end{matrix}$

where φ_(qb)=2 arctan (2χ/κ) is the qubit-induced phase of the outputfield. Moreover, the intracavity photon number is as

$\begin{matrix}{{\langle{{\hat{a}}^{\dagger}\hat{a}}\rangle} = {\left( \frac{2{\epsilon }}{\kappa} \right)^{2}{{{\cos^{2}\left( {\frac{1}{2}\phi_{qb}} \right)}\left\lbrack {1 - {2{\cos \left( {\chi \; t} \right)}e^{{- \frac{1}{2}}\kappa \; t}} + e^{{- \kappa}\; t}} \right\rbrack}.}}} & \left( {S\; 25} \right)\end{matrix}$

From the above expressions, the measurement signal is

$\begin{matrix}{{M_{S{1\rangle}} - M_{S{0\rangle}}} = {4{\epsilon }{\sin \left( \phi_{qb} \right)}{\sin \left( {\phi_{d} - \varphi_{h}} \right)}\tau \left\{ {1 - {\frac{4}{\kappa \; \tau}{{\cos^{2}\left( {\frac{1}{2}\phi_{qb}} \right)}\left\lbrack {1 - {\frac{\sin \left( {{\chi \; \tau} + \phi_{qb}} \right)}{\sin \left( \phi_{qb} \right)}e^{{- \frac{1}{2}}\kappa \; \tau}}} \right\rbrack}}} \right\}}} & ({S26})\end{matrix}$

On the other hand, the measurement noise is simply equal to

{circumflex over (M)}_(N) ²(T)

=κT. The measurement signal is optimized for

${\phi_{d} - \varphi_{h}} = {\frac{\pi}{2}{mod}\; \pi}$

and at long integration times by

${\phi_{qb} = \frac{\pi}{2}},$

or equivalently χ=κ/2. For this optimal choice, the SNR then reads

At long measurement times, the SNR evolves as

${SNR} = {\sqrt{8}\frac{\epsilon }{\kappa}\sqrt{\kappa \; \tau}}$

and at short measurement times it starts as:

$\begin{matrix}{{{SNR} = {\frac{1}{\sqrt{18}}\frac{\epsilon }{\kappa}\left( {\kappa \; \tau} \right)^{5/2}}}{{SNR} = {\sqrt{8}\frac{\epsilon }{\kappa}{{\sqrt{\kappa\tau}\left\lbrack {1 - {\frac{2}{\kappa\tau}\left( {1 - {e^{{- \frac{1}{2}}{\kappa\tau}}\cos \frac{1}{2}{\kappa\tau}}} \right)}} \right\rbrack}.}}}} & ({S27})\end{matrix}$

The presence of Purcell decay γ_(κ)=(g/Δ)²κ is taken into account usingthe expression of

{circumflex over (σ)}_(z)

(t) for a Purcell-limited qubit, i.e.,

{circumflex over (σ)}_(z)

(t)=(1+

{circumflex over (σ)}_(z)

(0))e^(−γκt)−1. The

$\begin{matrix}{{SNR} = {\sqrt{2}\frac{\epsilon }{\kappa}\sqrt{\kappa\tau}{\left\{ {1 - {\frac{2}{\kappa\tau}\left( {1 - {e^{{- \frac{1}{2}}{\kappa\tau}}\cos \frac{1}{2}{\kappa\tau}}} \right)} - {\frac{\kappa}{\tau}{\int_{0}^{\tau}{{dt}{\int_{0}^{t}{{dt}^{\prime}e^{{- \frac{1}{2}}{\kappa {({t - t^{\prime}})}}}{\sin\left\lbrack {{\frac{1}{2}{\kappa \left( {t - t^{\prime}} \right)}} - {\frac{\kappa}{\gamma_{\kappa}}\left( {e^{{- \gamma_{\kappa}}t^{\prime}} - e^{{- \gamma_{\kappa}}t}} \right)}}\  \right\rbrack}}}}}}} \right\}.}}} & ({S28})\end{matrix}$

corresponding SNR is then, for χ=κ/2,

D. Effect of a residual transverse coupling

In the fourth set of exemplary embodiments (D), presence of a spurioustransverse coupling g_(x) in addition to the longitudinal coupling g_(z)is considered whereby:

Ĥ=ω _(r) â ^(†) â+ω _(a){circumflex over (σ)}_(z) +{[g _(x) +{tilde over(g)} _(x) cos (ω_(r) t+φ _(x))]{circumflex over (σ)}_(x) +[g _(z)+{tilde over (g)} _(z) cos (ω_(r) t+φ)]{circumflex over (σ)}_(z)}(â ^(†)+â).   (S29)

It is now assumed that g_(x)«Δ and we follow the standard approach toeliminate the transverse coupling. To leading order in g_(x)/Δ and underthe RWA, the interaction picture is defined as:

{tilde over (H)}=½(χ_(x)−2χ_(xz)){circumflex over (σ)}_(z) +χâ ^(†)â{circumflex over (σ)} _(z)+½[{tilde over (g)} _(z) â ^(†) +{tilde over(g)} _(z) *â]{circumflex over (σ)} _(z),   (S30)

with the dispersive shifts χ=χ_(x)−4χ_(xz)·χ_(x)=g_(x) ⁻²/Δ and χ_(xz)=g_(x) g _(z)/Δ.

Going to an interaction picture also with respect to the first term ofEquation (S30), the starting point is

Ĥ=χâ ^(†) â{circumflex over (σ)} _(z)+½[{tilde over (g)} _(z) â ^(†)+{tilde over (g)} _(z) *â]{circumflex over (σ)} _(z).    (S31)

This leads to the Langevin equation

{circumflex over ({dot over (a)})}=−i½g _(z){circumflex over(σ)}_(z)−(iχ÷½κ)â−√κâ _(in).   (S32)

In accordance with previously presented results in (A), the measurementsignal is

$\begin{matrix}{{{\langle\hat{M}\rangle}_{1} - {\langle\hat{M}\rangle}_{0}} = {4{{\overset{\sim}{g}}_{z}}{\sin \left( {\phi - \varphi_{h}} \right)}\tau \; {\cos^{2}\left( {\frac{1}{2}\phi_{gb}} \right)}{\left\{ {1 - {\frac{2}{\kappa\tau}\left\lbrack {{\cos \left( \phi_{gb} \right)} - {{\cos \left( {\phi_{qb} + {\chi\tau}} \right)}e^{{- \frac{1}{2}}{\kappa\tau}}}} \right\rbrack}} \right\}.}}} & ({S33})\end{matrix}$

where as before it is noted that the dispersive-coupling-inducedrotation φ_(qb)=2 arctan (2χ/κ). Again as above, the measurement noiseis not changed by the dispersive shift. Choosing

${{\phi - \varphi_{h}} = {\frac{\pi}{2}{mod}\; \pi}},$

the SNR finally reads

$\begin{matrix}{{{SNR}(\chi)} = {\sqrt{8}\frac{{\overset{\sim}{g}}_{z}}{\kappa}\sqrt{\kappa\tau}{\cos^{2}\left( {\frac{1}{2}\phi_{qb}} \right)}{\left\{ {1 - {\frac{2}{\kappa\tau}\left\lbrack {{\cos \left( \phi_{qb} \right)} - {{\cos \left( {\phi_{qb} + {\chi\tau}} \right)}e^{{- \frac{1}{2}}{\kappa\tau}}}} \right\rbrack}} \right\}.}}} & ({S34})\end{matrix}$

The residual dispersive coupling reduces the value of the SNR, with thedecrease behaving differently at long and short measurement times. Atlong measurement times, the dispersive coupling reduces the SNR by

$\begin{matrix}{{{{{SNR}(\chi)} \simeq {{\cos^{2}\left( {\frac{1}{2}\phi_{qb}} \right)}{{SNR}\left( {\chi = 0} \right)}}} = {\frac{\kappa^{2}}{\kappa^{2} + {4\chi^{2}}}{{SNR}\left( {\chi = 0} \right)}}},{{{for}\mspace{14mu} \tau}{1/{\kappa.}}}} & ({S35})\end{matrix}$

The SNR is not affected for χ«κ/2. Interestingly, at short measurementtimes the SNR is completely independent of the spurious dispersive shiftto leading orders

$\begin{matrix}{{{{SNR}(\chi)} \simeq {\frac{1}{\sqrt{2}}\frac{{\overset{\sim}{g}}_{z}}{\kappa}\left( {\kappa \; \tau} \right)^{3/2}\left( {1 - {\frac{1}{6}\kappa \; \tau}} \right)}},{{{for}\mspace{14mu} \tau}{1/{\kappa.}}}} & \left( {S\; 36} \right)\end{matrix}$

In short, the SNR is not affected by a spurious transverse coupling forshort measurement times τ«1/κ.

II. Circuit QED Realization

An exemplary realization of longitudinal coupling in circuit QED is nowaddressed. While a lumped circuit QED implementation 700 is presented inFIG. 7, focus is on put a transmon qubit that is phase-biased by acoplanar waveguide resonator. The lumped element results are recoveredin the appropriate limit. Emphasis is put on numerical results obtainedfor this coplanar realization. For this reason these numerical valuesdiffer from, but are compatible with, what is found with reference toFIGS. 1 to 9.

As illustrated in FIG. 10, a transmon qubit (110) coupled at middle ofthe center conductor of a λ/2 resonator (109). To increase thelongitudinal coupling strength, a Josephson junction can be inserted inthe center conductor of the resonator at the location of the qubit(111). FIG. 10 shows a logical representation of physicalcharacteristics of a transmon qubit coupling longitudinally to aresonator. The coplanar waveguide resonator is composed of a centralelectrode (109) surrounded by a ground plane (112). A couplinginductance (111) mediates the longitudinal coupling to the transmonqubit (110). A capacitively coupled transmission line (113) allows tosend and retrieve input and output signals to and from the resonator.FIG. 11 shows a close-up of the transmon qubit with the definitions ofthe branch fluxes used herein. The qubit is composed of nominallyidentical Josephson junctions (114) and large shunting capacitances(115). Additional control lines are needed to modulate the flux throughthe loop φ_(x) and to perform single-qubit operations (not shown on FIG.10 and FIG. 11). Ultrastrong transverse coupling of a flux qubit to aresonator has been presented by Bourassa, J. et. al. in “Ultrastrongcoupling regime of cavity QED with phase-biased flux qubits”, whichinvolve at least some inventors also involved in an invention claimedherein. The modelling of the present circuit closely follows Bourassa,J. et. al. and relevant details are provided herein for completeness.

The Lagrangian of this circuit,

=

r+

q+

qr, is composed of three parts consisting of the bare resonator

r, qubit

q and interaction

qr Lagrangians. From standard quantum circuit theory, the resonatorLagrangian takes the form

$\begin{matrix}{{\left( \frac{2\; \pi}{\Phi_{0}} \right)^{2}\mathcal{L}_{r}} = {{\int_{{- L}/2}^{L/2}{\left( {{\frac{C^{0}}{2}{{\overset{.}{\psi}}^{2}\left( {x,t} \right)}} - {\frac{1}{2L^{0}}\left( {\partial_{x}{\psi \left( {x,t} \right)}} \right)^{2}}} \right){dx}}} + {\left( \frac{2\; \pi}{\Phi_{0}} \right)^{2}E_{Jr}{\cos \left\lbrack {\Delta \; \psi} \right\rbrack}} + {\frac{C_{Jr}}{2}\overset{.}{\Delta}\; {\psi.}}}} & \left( {S\; 37} \right)\end{matrix}$

where ψ(x) is the position-dependant field amplitude inside theresonator and Δψ=ψ(x_(a)+Δx/2)−ψ(x_(a)+Δx/2) is the phase bias acrossthe junction of width Δx at position x_(a). In this expression, it isassumed that the resonator has total length L with capacitance C⁰ andinductance L⁰ per unit length. In the single mode limit, ψ(x,t)=ψ(t)u(x) where u(x) is the mode envelope. The Josephson junction inthe resonator's center conductor has energy E_(Jr) and capacitanceC_(Jr). This junction creates a discontinuity Δψ≈0 in the resonatorfield that will provide the desired longitudinal interaction. Thecoupling inductance can be replaced by a SQUID, or SQUID array, withoutsignificant change to the treatment. This Lagrangian was already studiedin the context of strong transverse coupling flux qubits totransmission-line resonators and for non-linear resonators.

The transmon qubit is composed of a capacitor to ground C_(b) and twocapacitively shunted Josephson junctions of energies E_(J1) and E_(J2),and total capacitances C_(q1)=C_(j1)+C_(s1) and C_(q2)=C_(j2)+C_(s2)respectively. In terms of the branch fluxes defined on FIG. 11, thequbit Lagrangian taking into account the coupling to the resonator is

$\begin{matrix}{{\mathcal{L}_{q} + \mathcal{L}_{qr}} = {{\left( \frac{\Phi_{0}}{2\; \pi} \right)^{2}\left\lbrack {{\frac{C_{q\; 1}}{2}\left( {{\overset{.}{\psi}}_{1} - \overset{.}{\varphi} - {\overset{.}{\Phi}}_{x}} \right)^{2}} + {\frac{C_{q\; 2}}{2}\left( {{\overset{.}{\psi}}_{2} - \overset{.}{\varphi}} \right)^{2}} + {\frac{C_{b}}{2}{\overset{.}{\varphi}}^{2}}} \right\rbrack} + {E_{J\; 1}{\cos \left\lbrack {\psi_{1} - \varphi_{1} - \Phi_{x}} \right\rbrack}} + {E_{J\; 2}{\cos \left\lbrack \varphi_{2} \right\rbrack}}}} & \left( {S\; 38} \right)\end{matrix}$

Here ψ₁₍₂₎=ψ(x_(a)∓Δ_(x)/2) is defined for simplicity and the qubitcapacitances C_(q)=C_(j)+C_(s) for each arm. Defining new variablesθ=(ψ₁+ψ₂−2φ)/2 and δ=(ψ₁+ψ₂+2φ)/2, the above is obtained

$\begin{matrix}{{\mathcal{L}_{q} + \mathcal{L}_{qr}} = {\left( \frac{\Phi_{0}}{2\; \pi} \right)^{2}{\quad{\left\lbrack {{\frac{C_{q\; 1} + C_{q\; 2}}{2}\left( {{\overset{.}{\theta}}^{2} + {\overset{.}{\Delta}{\psi^{2}/4}}} \right)} + {\frac{C_{b}}{8}\left( {\overset{.}{\delta} - \overset{.}{\theta}} \right)^{2}} + {\frac{\left( {C_{q\; 1} - C_{q\; 2}} \right)}{2}\overset{.}{\theta}\overset{.}{\Delta}\psi}} \right\rbrack + {E_{J\; 1}{\cos \left\lbrack {\theta - {\Delta \; {\psi/2}} + \Phi_{x}} \right\rbrack}} + {E_{J\; 2}{{\cos \left\lbrack {\theta + {\Delta \; {\psi/2}}} \right\rbrack}.}}}}}} & \left( {S\; 39} \right)\end{matrix}$

Shifting the variable θ→θ+φ_(x)/2, a more symmetrical Lagrangian isobtained with respect to the external flux

$\begin{matrix}{{\mathcal{L}_{q} + \mathcal{L}_{qr}} = {{\left( \frac{\Phi_{0}}{2\; \pi} \right)^{2}\left\lbrack {{\frac{C_{q\; 1} + C_{q\; 2}}{2}\left( {{\overset{.}{\theta}}^{2} + \frac{{\Delta \; {\overset{.}{\psi}}^{2}} + {\overset{.}{\Phi}}_{x}^{2}}{4}} \right)} + {\frac{C_{b}}{8}\left( {\overset{.}{\delta} - \overset{.}{\theta} - {{\overset{.}{\Phi}}_{x}/2}} \right)^{2}} + {\frac{\left( {C_{q\; 1} - C_{q\; 2}} \right)}{2}{\overset{.}{\theta}\left( {{\overset{.}{\Phi}}_{x} - {\overset{.}{\Delta}\psi}} \right)}} + {\frac{C_{q\; 1} + C_{q\; 2}}{4}\overset{.}{\Delta}\; \psi \; {\overset{.}{\Phi}}_{x}}} \right\rbrack} + {E_{J\; 1}{\cos \left\lbrack {\theta - {\Delta \; {\psi/2}} - {\Phi_{x}/2}} \right\rbrack}} + {E_{J\; 2}{{\cos \left\lbrack {\theta + {\Delta \; {\psi/2}} + {\Phi_{x}/2}} \right\rbrack}.}}}} & \left( {S\; 40} \right)\end{matrix}$

The equation of motion for δ being {umlaut over (δ)}−{umlaut over(θ)}+{umlaut over (Φ)}_(x)/2=0, the term involving the capacitance toground C_(b) is thus a constant and can be ignored. Assuming theresonator phase-bias to be small Δψ«1 and a DC-flux bias ({dot over(Φ)}_(r)=0), we find to zeroth-order in Δψ the usual Lagrangian of anasymmetric fluxbiased transmon qubit

$\begin{matrix}{{\mathcal{L}_{q} = {{\left( \frac{\Phi_{0}}{2\; \pi} \right)^{2}\frac{C_{q\; 1} + C_{q\; 2}}{2}{\overset{.}{\theta}}^{2}} + {E_{J\; \Sigma}\left\lbrack {{{\cos \left( {\Phi_{x}/2} \right)}\cos \; \theta} - {d\; {\sin \left( {\Phi_{x}/2} \right)}\sin \; \theta}} \right\rbrack}}},} & \left( {S\; 41} \right)\end{matrix}$

where E_(JΣ)=E_(J1)+E_(J2) and d=(E_(J2)−E_(J1))/E_(JΣ) is the junctionasymmetry. Defining {circumflex over (n)} to be the conjugate charge to{circumflex over (θ)}, the corresponding Hamiltonian is

Ĥ _(q) =ΛE _(C){circumflex over (n)}²−E_(JΣ)[cos (Φ_(x)/2) cos{circumflex over (θ)}−d sin (Φ_(x)/2) sin {circumflex over (θ)}].  (S12)

with the charging energy E_(C)=e²/2(C_(q1)+C_(q2)).

For clarity, projection is made on the qubit subspace {|0>, |1>} wherethe total Hamiltonian takes the form

$\begin{matrix}{\hat{H} = {{\omega_{r}{\hat{a}}^{\dagger}\hat{a}} + {K\left( {{\hat{a}}^{\dagger}\hat{a}} \right)}^{2} + {\frac{\omega_{a}}{2}{\hat{\sigma}}_{z}} + {{\hat{H}}_{qr}.}}} & \left( {S\; 43} \right)\end{matrix}$

In this expression, ω_(a) is the qubit transition frequency and K theKerr non-linearity. The latter can be made small and in particularnegligible with respect to the photon decay rate κ. To first order inΔ{circumflex over (ψ)} and in the same two-level approximation, theinteraction Hamiltonian reads

$\begin{matrix}\begin{matrix}{{\hat{H}}_{qr} = {E_{J\; \Sigma}{\frac{\Delta \; \hat{\psi}}{2}\left\lbrack {{{\sin \left( {\Phi_{x}/2} \right)}\cos \; \hat{\theta}} + {d\; {\cos \left( {\Phi_{x}/2} \right)}\sin \; \hat{\theta}}} \right\rbrack}}} \\{{= {{{g_{z}\left( {{\hat{a}}^{\dagger} + \hat{a}} \right)}{\hat{\sigma}}_{z}} + {{g_{x}\left( {{\hat{a}}^{\dagger} + \hat{a}} \right)}{\hat{\sigma}}_{x}}}},}\end{matrix} & \left( {S\; 44} \right)\end{matrix}$

where

$\begin{matrix}{{{g_{z}\left( \Phi_{x} \right)} = {{- \psi_{rms}}\Delta \; u{\frac{E_{J\; \Sigma}}{4}\left\lbrack {{m_{11}\left( \Phi_{x} \right)} - {m_{00}\left( \Phi_{x} \right)}} \right\rbrack}}},} & \left( {S\; 45} \right) \\{{g_{x}\left( \Phi_{x} \right)} = {\psi_{rms}\Delta \; u\frac{E_{J\; \Sigma}}{2}{{m_{01}\left( \Phi_{x} \right)}.}}} & \left( {S\; 46} \right)\end{matrix}$

To obtain these expressions, Δ{circumflex over (ψ)}=Δuψ_(rms)(â^(†)+â)is used, with ψ_(rms)=√4πZ_(r)/R_(K). Here Z_(r) is the resonator modeimpedance and R_(K)=h/e² the quantum of resistance. The mode gap Δu isfound to be optimal for a junction placed in the center of the resonatorcenter (see Bourassa, J. et. al.). At that location, Δu can be relatedto the participation ratio of the coupling inductance L_(J)⁻¹=E_(Jr)(2π/φ₀)² to the effective inductance of the resonator modeL_(r) as Δu≈2η=2L_(J)/L_(r) The factor of two originates from the factthat both resonator halves bias equally the qubit. The lumped elementlimit discussed in the previous discussion is obtained for η→1/2.Moreover, it has been defined that m_(ij)=

i| sin (Φ_(x)/2) cos {circumflex over (θ)}+d cos (Φ_(x)/2) sin{circumflex over (θ)}|j

. Equations (S46) and (S45) are calculated numerically by diagonalizingthe transmon Hamiltonian.

B. Numerical evaluation of the coupling strength

A set of possible parameters is presented for this circuit. It is veryimportant to emphasize that these numbers relate to the coplanararchitecture discussed here and not to the lumped-element exampledescribed in relation to FIGS. 7 to 9. A λ/2 resonator is used ofcharacteristic impedance Z⁰˜50Ω, total length 7.6 mm and capacitance perunit length C⁰=0.111 nF/m and inductance per unit length L⁰=0.278 μH/m.The coupling inductance consists of an array of 13 Josephson Junctionswith a total Josephson energy E_(Jr)/h=420 GHz. Following the standardprocedure, the first resonator mode is determined to be at frequencyω_(r)/2π=10.02 GHz, with a characteristic mode impedance Z_(r)=18.8Ω.The participation ratio of the array is found to be η˜0.3. With theseparameters, the resonator bias on the qubit is ψ_(rms)≈0.096.

The transmon Josephson and charging energies are E_(JΣ)/h=20 GHz andE_(c)/h=0.3 GHz respectively, and the asymmetry d=0.02. These parametersyield ω_(a)/2π=6.6 GHz when evaluated at the flux sweet-spot Φ_(x)=0.The magnetic flux is modulated around this flux value with a maximumexcursion of Φ_(z)/2π=0.1. At the sweet-spot, the qubit-resonator isΔ=ω_(n)(0)−ω_(r)≈2π×3.4 GHz The longitudinal coupling {tilde over(g)}_(z) the various spurious interactions found numerically using thesenumbers are summarized in Table I, which provides longitudinal couplingrate {tilde over (g)}_(z) and spurious leading-order and second-ordercouplings for a transmon coupled to a λ/2 resonator with a couplinginductance consisting of an array of 13 Josephson junctions as discussedherein above with reference to the coplanar architecture (i.e., not inrelation to the lumped element version discussion in FIGS. 7 to 9).

Rates Couplings (2π × MHz) Longitudinal [Eq. (S48)] {tilde over (g)}_(z)26.3 Transverse [Eq. (S49)] g_(x) 7.89 Dispersive g_(x) ²/Δ 0.018Non-linear dispersive [Eq. (S56)] 2Λ _(z) 3.97 Non-linear transverse[Eq. (S57)] Λ _(x) 0 Kerr nonlinearity K 0.007 Qubit frequency shift(from flux-drive) [Eq. (S54)] ϵ_(q) ²/Δ 4.4 × 10⁻⁵

In particular, {tilde over (g)}_(z)/2π=26.3 MHz and a small transversecoupling resulting in a negligible dispersive shift χ_(x)=g_(x) ²/Δ˜18kHz. The maximum change in qubit frequency is

ω_(a)(0)−ω_(a)({tilde over (Φ)})˜2π×170 MHz.

The array of N junctions allows for strong longitudinal coupling whilereducing the resonator nonlinearity. Compared to a single junction ofequal energy, the non-linear Kerr effectively decreases as K=K₁/N² andfrom the parameters above K/2π=7 kHz using 13 junctions is obtained.

C. Asymptotic expression for the coupling strengths

To obtain asymptotic expressions for the qubit-resonator coupling, thetransmon is considered as a weakly anharmonic oscillator. In thissituation,

$\begin{matrix}{{\overset{\sim}{\theta} \approx {\left\lbrack \frac{2\; E_{C}}{E_{J}\left( \Phi_{s} \right)} \right\rbrack^{1/4}\left( {{\hat{b}}^{\dagger} + \hat{b}} \right)\mspace{14mu} {and}}}\mspace{14mu} {{n_{i,{j \neq i}} \approx {{d\left( \frac{2\; E_{C}}{E_{J}\left( \Phi_{x} \right)} \right)}^{1/4}{\cos \left( {\Phi_{x}/2} \right)}{\langle{i{\left( {{\hat{b}}^{\dagger} + \hat{b}} \right)}j}\rangle}}},{m_{ii} \approx {{- \left( \frac{2\; E_{C}}{E_{J}\left( \Phi_{x} \right)} \right)^{1/2}}\frac{\sin \left( {\Phi_{x}/2} \right)}{2}{\langle{i{\left( {{\hat{b}}^{\dagger} + \hat{b}} \right)^{2}}i}\rangle}}}}} & \left( {S\; 47} \right)\end{matrix}$

Restricting to the {|0

, |1

} subspace and using Equations (S46) and (S45), the asymptoticexpressions obtained are

$\begin{matrix}{{g_{z} \approx {{- {\frac{E_{J\; \Sigma}}{2}\left\lbrack \frac{2\; E_{C}}{E_{J}\left( \Phi_{x} \right)} \right\rbrack}^{1/2}}\sqrt{\frac{\pi \; Z_{r}}{R_{K}}}{\sin \left( {\Phi_{x}/2} \right)}2\; \eta}},} & \left( {S\; 48} \right) \\{g_{x} \approx {{{dE}_{J\; \Sigma}\left\lbrack \frac{2\; E_{C}}{E_{J}\left( \Phi_{x} \right)} \right\rbrack}^{1/4}\sqrt{\frac{\pi \; Z_{r}}{R_{K}}}{\cos \left( {\Phi_{x}/2} \right)}2\; {\eta.}}} & \left( {S\; 49} \right)\end{matrix}$

These correspond to Equations 5 and 6 previously discussed where η→1/2.

D. Upper bounds on the coupling strength

An upper bounds for the longitudinal coupling is obtained by expressingg_(z) in units of the qubit frequency. Using Equation S48:

$\begin{matrix}{\frac{g_{z}}{\omega_{a}} \approx {\frac{\Phi_{x\;}}{8}\sqrt{\frac{\pi \; Z_{r}}{R_{K}}}2\; {\eta.}}} & \left( {S\; 50} \right)\end{matrix}$

For Z_(r)˜50Ω resonator, maximal coupling is reached in thelumped-element limit of the oscillator where η→1/2 and we getg_(z)/ω_(a)˜0.006, or about twice the coupling obtained in the previousexample giving g_(z)/2π˜42.4 MHz.

Even larger values of g_(z) can be achieved by using lumped LC-circuitcomprises of a superinductance of large impedance Z_(r)˜R_(K)/4. Forparticipation ratios in the range 2η˜[10⁻², 1], the coupling is enhancedby a factor of ˜10 to g_(z)/ω_(a)=Φ×η√π/8˜[7×10⁻⁴, 7×10⁻²]. While theprevious circuit model would have to be redefined to take the large biasΔψ into account, it is safe to say that the larger and more compactcoupling inductance is, the stronger the longitudinal coupling will be.

III. Spurious Couplings and Imperfections

From the first term in Equation (S39), an oscillating external magneticflux Φ_(x)(t)={tilde over (Φ)}_(x) cos (ω_(r)t+φ) at frequency ω_(d)leads to an effective voltage driver on the resonator

$\begin{matrix}{\mathcal{L}_{r.{drive}} = {\left( \frac{\Phi_{0}}{2\; \pi} \right)^{2}\frac{C_{q\; 1} + C_{q\; 2}}{2}{\overset{.}{\Phi}}_{x}{{\overset{.}{\psi}(0)}.}}} & \left( {S\; 51} \right) \\{{\epsilon_{r} = {\frac{h\; \omega_{r}}{16\; E_{CJ}}\psi_{rms}\Phi_{x}\omega_{d}}},} & \left( {S\; 52} \right)\end{matrix}$

This term leads to an effective driver on the resonatorĤ_(r,d)=ϵ_(r)e^(−i(ω) ^(r) ^(t+φ))â^(†)+H.c. of drive amplitude ϵ_(r)

where E_(CJ)=e²/[2(C_(q1)+C_(q2))]. With the above circuit, parameters,ϵ_(r)/2π˜5 MHz is obtained. If desired, the effect of this drive can becancelled by an additional drive on the input port of the resonator.Otherwise, this simply leads to an additional qbit-state independentdisplacement of the cavity field that does not affect the SNR.

In the same way, flux modulation also directly drives the qubit. This iscaused by the last charging energy term of Equation (S39) and yields

$\begin{matrix}{{{\hat{H}}_{q,d} = {\left( \frac{\Phi_{0}}{2\; \pi} \right)^{2}\frac{C_{q\; 1} - C_{q\; 2}}{2}{\overset{.}{\Phi}}_{x}\hat{q}}},} & \left( {S\; 53} \right)\end{matrix}$

From the asymptotic expression {circumflex over(q)}≈i(E_(J)/32E_(C))^(1/4)(b^(†)−b), Ĥ_(q,d)=ϵ_(q)e^(−k(ω) ^(r)^(t)+φ){circumflex over (b)}^(†)+H.C. is obtained where

$\begin{matrix}{\epsilon_{q} \approx {\frac{d}{8}\frac{C_{J}}{C_{\Sigma}}\left( \frac{E_{J}}{32E_{C}} \right)^{1/4}\Phi_{x}\omega_{r}}} & \left( {S\; 54} \right)\end{matrix}$

is identically zero for symmetric junctions (d=0). With the aboveparameters and C_(J)/C_(Σ)=0.01, then ϵ_(q)/2π˜0.18 MHz. Given thestrong qubit-resonator detuning of several GHz, this is however of noconsequences.

B. Higher-order interaction terms

To second order in Δψ in Equation (S39) yields the additionalinteractions

Ĥ _(qr) ⁽²⁾=(â ^(†) +â)²(Λ_(x){circumflex over (σ)}_(x)+Λ_(z){circumflexover (σ)}_(z)).   (S55)

where

$\begin{matrix}{{\Lambda_{z} = {\psi_{rms}^{2}\frac{E_{J\sum}}{16}\left( {{{\langle 0}\hat{cc}} - {d\mspace{14mu} {\hat{ss}\left\lbrack 0\rangle \right.}} - {{\langle 1}\hat{cc}} - {d\mspace{14mu} \hat{ss}{1\rangle}}} \right)}},} & ({S56}) \\{\Lambda_{x} = {\psi_{rms}^{2}\frac{E_{J\sum}}{8}{{\langle 1}\left\lbrack {\hat{cc} - {d\mspace{14mu} \hat{ss}}} \right\rbrack}{{0\rangle}.}}} & ({S57})\end{matrix}$

and where have been defined ĉc=cos (Φ_(x)/2) cos {circumflex over (θ)}and ŝs=sin (Φ_(x)/2) sin {circumflex over (θ)}.

Under flux modulation, Λ_(k)(t)=Λ_(k)+{tilde over (Λ)}_(k) cos(ω_(r)t+φ_(r)) with k=x,z. With the RWA this leads to

Ĥ _(qr) ⁽²⁾≈(2â ^(†) â+1)(Λ_(x){circumflex over(σ)}_(x)+Λ_(x){circumflex over (σ)}_(z)).   (S58)

For d small and flux modulations around Φ_(x)=0 then Λ _(x)=0. On theother hand, the dispersive-like interaction of amplitude χ_(z)=2Λ _(z)has a magnitude of χ_(z)/2π≈3.97 MHz for the above circuit parameters.In the lumped-element limit with Z_(r)=50Ω, the result is χ_(z)/2π=5.3MHz. Since this dispersive-like interaction originates from asecond-order correction, it is always found that g_(z)/χ_(z)≈10. Inpractice, choosing χ_(z)<κ will allow this spurious coupling will notaffect the SNR at short integration times.

IV. Multiple Qubit Read-Out in (X+Z) Circuit QED

FIG. 12 illustrates a possible multi-qubit architecture: qubits (110)are longitudinally coupled to a readout resonator (109) andtransversally coupled to a bus resonator (120). The readout resonator isused only when the longitudinal couplings are modulated and individualflux control for each qubit is possible with separated flux lines. Thesystem Hamiltonian reads

$\begin{matrix}{\hat{H} = {{\omega_{rz}a_{z}^{\dagger}{\hat{a}}_{z}} + {\omega_{rx}{\hat{a}}_{x}^{\dagger}{\hat{a}}_{x}} + {\sum\limits_{j}{g_{zj}{{\hat{\sigma}}_{zj}\left( {{\hat{a}}_{z}^{\dagger} + {\hat{a}}_{z}} \right)}}} + {\sum\limits_{j}{g_{xj}{{{\hat{\sigma}}_{xj}\left( {{\hat{a}}_{x}^{\dagger} + {\hat{a}}_{x}} \right)}.}}}}} & ({S59})\end{matrix}$

In the interaction picture and neglecting fast-oscillating terms, thelongitudinal coupling becomes

$\begin{matrix}{{\overset{\sim}{H}}_{z} = {{\left( {\frac{1}{2}{\overset{\sim}{g}}_{z}{\sum\limits_{j}{{\hat{\sigma}}_{zj}e^{i_{\phi,i}}}}} \right){\hat{a}}_{z}} + {H.c.}}} & ({S60})\end{matrix}$

The position of the qubits and the coupling inductances can be adjustedto get equal longitudinal coupling strengths |g_(zj)|. By properlychoosing the phases φ_(j), it is possible to operate in the joint qubitreadout mode or in the entanglement by measurement mode. Examples for 2and 3 qubits are shown in FIGS. 13A, 13B, 13C and 13D (joint measurementof two qubits, synthesis of Bell states and 3-qubit GHZ state bymeasurement).

FIG. 12 shows a multi-qubit architecture consisting of qubits (110)longitudinally coupled to a readout resonator (109) and transversallycoupled to a bus resonator (120) by way of bus-resonator fingers (119)capacitively coupled to the qubits. As in FIG. 10, the read-outresonator is coupled to an input transmission-line (113) but also to andoutput transmission-line (114). Similarly, the bus resonator isconnected to both an input (116) and output (117) transmission-line forcontrol. The ground plane (112) is placed on both sides of theresonators. Between the qubits (110), the pieces of ground planes areinterconnected with air bridges (118). Flux control lines and qubitcontrols lines for each qubit are absent in this figure. FIGS. 13A, 13B,13C and 13D show phase space representation of the pointer states fortwo and three qubits. On FIG. 13A, two-qubit joint readout (φ_(j)=jπ/2).On FIGS. 13B and 13C, two-qubit entanglement by measurement is depicted.The double-hatched circles centered at the origin circles correspond tothe Bell states

${\Psi^{-}\rangle} = {\frac{1}{\sqrt{2}}\left\lbrack {{{0,1}\rangle} + {{1,0}\rangle}} \right\rbrack}$

and

${{\Phi^{-}\rangle} = {\frac{1}{\sqrt{2}}\left\lbrack {{{0,0}\rangle} + {{1,1}\rangle}} \right\rbrack}},$

obtained for φ_(j)=0 and φ_(j)=jπ, respectively. On FIG. 13D, thecentered at the origin circle corresponds to the GHZ state

${{GHZ}\rangle} = {{\frac{1}{\sqrt{2}}\left\lbrack {{{0,0,0}\rangle} + {{1,1,1}\rangle}} \right\rbrack}.}$

This configuration is obtained for φ_(j)=j2π/3

V. Qubit Dephasing

When not measuring the qubit, coupling to the resonator is turned offand the qubit is not dephased by photon shot noise or residual thermalpopulation. In the superconducting qubit architecture studied in theprevious section, another potential source of dephasing is flux noise.The influence of low-frequency flux noise on the transmon was alreadystudied. Since the qubit is operated at its flux sweet spot, the effectof this noise can be negligible.

More interesting here is the contribution opened by the longitudinalcoupling, that is flux noise at the resonator frequency which willeffectively ‘measure’ the qubit. Following the standard procedure andusing Equation (5) previously defined, in the presence of flux noise,longitudinal coupling leads to a decay of the qubit's off-diagonaldensity matrix element with

$\begin{matrix}{{\rho_{0\; 1}(t)} = {{\rho_{01}(0)}e^{i\; \omega_{n}t}\exp {\left\{ {\frac{- \lambda^{2}}{2}\frac{\omega}{2\pi}{{S_{\Phi}\left( {\omega + \omega_{r}} \right)}\left\lbrack {{\left( {{2\overset{\_}{n}} + 1} \right)\frac{\sin^{2}\left( {\omega \; {t/2}} \right)}{\omega^{2}}} + {2\overset{\_}{m}\mspace{14mu} {\cos \left( {{\omega_{r}t} - \theta} \right)}\frac{{\sin \left( {\omega \; {t/2}} \right)}{\sin \left( {\left( {\omega + {2\omega_{r}}} \right){t/2}} \right)}}{\omega \left( {\omega + {2\omega_{r}}} \right)}}} \right\rbrack}} \right\}.}}} & \left( {S\; 61} \right)\end{matrix}$

In this expression, S₁₀₁ (ω) is the spectral density of flux noise, n=

â^(†)â

, me^(iθ)=

â²

and

$\begin{matrix}{\lambda = {\left. \frac{\partial g_{z}}{\partial\Phi_{x}} \right|_{\Phi_{x} = 0} = {{- \frac{\pi}{\Phi_{0}}}\frac{E_{J}}{2}\left( \frac{2E_{C}}{E_{J}} \right)^{1/2}{\sqrt{\frac{\pi \; Z_{0}}{R_{K}}}.}}}} & ({S62})\end{matrix}$

The term inρ₀₁(t) not proportional to n or m takes the standard form,with the crucial difference that here the noise spectral density isevaluated at ω+ω_(r) rather than at ω. With

S_(Φ)(ω)=2πA²/|ω|^(a) even at large frequencies, dephasing caused bylongitudinal coupling is therefore expected to be negligible. Inpractice, the terms proportional to n or m are expected to contributeonly in the situation where the intra-cavity field would be squeezedprior to the qubit measurement. In this situation, n=sin h²r and m=−sinh(2r)/2 with r the squeeze parameter.

Although decay is not exponential, the dephasing time can be estimatedfrom the above expression by using the 1/e threshold as an estimate. Forthis an infrared cutoff must be introduced. Fortunately, the end resultis only weakly dependent on this cutoff. For simplicity α=1 is taken inthe noise spectral density with A=10⁻⁵Φ₀. Using the parameters given inthe text, in the absence of squeezing but with a large spurious photonnumber n=0.1 a very large dephasing time of ˜10² seconds is found. For asqueezed state of 20 dB a dephasing time larger than a second is found.This mechanism is clearly not limiting the qubit dephasing time.

A method is generally conceived to be a self-consistent sequence ofsteps leading to a desired result. These steps require physicalmanipulations of physical quantities. Usually, though not necessarily,these quantities take the form of electrical or magnetic/electromagnetic signals capable of being stored, transferred, combined,compared, and otherwise manipulated. It is convenient at times,principally for reasons of common usage, to refer to these signals asbits, values, parameters, items, elements, objects, symbols, characters,terms, numbers, or the like. It should be noted, however, that all ofthese terms and similar terms are to be associated with the appropriatephysical quantities and are merely convenient labels applied to thesequantities. The description of the present invention has been presentedfor purposes of illustration but is not intended to be exhaustive orlimited to the disclosed embodiments. Many modifications and variationswill be apparent to those of ordinary skill in the art. The embodimentswere chosen to explain the principles of the invention and its practicalapplications and to enable others of ordinary skill in the art tounderstand the invention in order to implement various embodiments withvarious modifications as might be suited to other contemplated uses.

-   -   We investigate an approach to universal quantum computation        based on the modulation of longitudinal qubit-oscillator        coupling. We show how to realize a controlled-phase gate by        simultaneously modulating the longitudinal coupling of two        qubits to a common oscillator mode. In contrast to the more        familiar transversal qubit-oscillator coupling, the magnitude of        the effective qubit-qubit interaction does not rely on a small        perturbative parameter. As a result, this effective interaction        strength can be made large, leading to short gate times and high        gate fidelities. We moreover show how the gate fidelity can be        exponentially improved with squeezing and how the entangling        gate can be generalized to qubits coupled to separate        oscillators. Our proposal can be realized in multiple physical        platforms for quantum computing, including superconducting and        spin qubits.

Introduction.—A widespread strategy for quantum information processingis to couple the dipole moment of multiple qubits to common oscillatormodes, the latter being used to measure the qubits and to mediatelong-range interactions. Realizations of this idea are found in Rydbergatoms, superconducting qubits and quantum dots amongst others. With thedipole moment operator being off-diagonal in the qubit's eigenbasis,this type of transversal qubit-oscillator coupling leads tohybridization of the qubit and oscillator degrees of freedom. In turn,this results in qubit Purcell decay and to qubit readout that is nottruly quantum non-demolition (QND). To minimize these problems, thequbit can be operated at a frequency detuning from the oscillator thatis large with respect to the transverse coupling strength g_(x). Thisinteraction then only acts perturbatively, taking a dispersivecharacter. While it has advantages, this perturbative character alsoresults in slow oscillator-mediated qubit entangling gates.

Rather than relying on the standard transversal coupling,H_(x)=g_(z)(â^(†)+â){circumflex over (σ)}_(x), an alternative approachis to use a longitudinal interaction, H_(z)=g_(z)(â^(†)+â){circumflexover (σ)}_(z). Since H_(z) commutes with the qubit's bare Hamiltonianthe qubit is not dressed by the oscillator. Purcell decay is thereforeabsent and qubit readout is truly QND. The absence of qubit dressingalso allows for scaling up to a lattice of arbitrary size with strictlylocal interactions.

By itself, longitudinal interaction however only leads to a vanishinglysmall qubit state-dependent displacement of the oscillator field ofamplitude g_(z)/ω_(r)«1, with ω_(r) the oscillator frequency. In PaperA, it was shown that modulating g_(z) at the oscillator frequency ω_(r)activates this interaction leading to a large qubit state-dependentoscillator displacement and to fast QND qubit readout. In this Letter,we show how the same approach can be used, together with single qubitrotations, for universal quantum computing by introducing a fast andhigh-fidelity controlled-phase gate based on longitudinal coupling. Thetwo-qubit logical operation relies on parametric modulation of alongitudinal qubit-oscillator coupling, inducing an effective{circumflex over (σ)}_(z){circumflex over (σ)}_(z) interaction betweenqubits coupled to the same mode. We show that, with a purelylongitudinal interaction, the gate fidelity can be improvedexponentially using squeezing, and that the gate can be performedremotely on qubits coupled to separate but interacting oscillators. Thelatter allows for a modular architecture that relaxes design constraintsand avoids spurious interactions while maintaining minimal circuitcomplexity.

In contrast to two-qubit gates based on a transversal interaction, thisproposal does not rely on strong qubit-oscillator detuning and is notbased on a perturbative argument. As a result, the longitudinallymediated {circumflex over (σ)}_(z){circumflex over (σ)}_(z) interactionis valid for all qubit, oscillator and modulation parameters and doesnot result in unwanted residual terms in the Hamiltonian. For thisreason, in the ideal case where the interaction is purely longitudinal(i.e. described by H_(z)), there are no fundamental bounds on gateinfidelity or gate time and both can in principle be made arbitrarilysmall simultaneously.

Similarly to other oscillator-mediated gates, loss from the oscillatorduring the gate leads to gate infidelity. This can be minimized byworking with high-Q oscillators something that is, however, incontradiction with the requirements for fast qubit readout. We solvethis dilemma by exploiting quantum bath engineering, using squeezing atthe oscillator input. By appropriately choosing the squeezed quadrature,we show how ‘which-qubit-state’ information carried by the photonsleaving the oscillator can be erased. This leads to an exponentialimprovement in gate fidelity with squeezing strength.

Oscillator mediated qubit-qubit interaction.—Following Paper A, weconsider two qubits coupled to a single harmonic mode via their{circumflex over (σ)}_(z) degree of freedom. Allowing for atime-dependent coupling, the Hamiltonian reads (h=1)

$\begin{matrix}{{\hat{H}(t)} = {{\omega_{r}{\hat{a}}^{\dagger}\hat{a}} + {\frac{1}{2}\omega_{a\; 1}{\hat{\sigma}}_{z\; 1}} + {\frac{1}{2}\omega_{a\; 2}{\hat{\sigma}}_{z\; 2}} + {{g_{1}(t)}{{\hat{\sigma}}_{z\; 1}\left( {{\hat{a}}^{\dagger} + \hat{a}} \right)}} + {{g_{2}(t)}{{{\hat{\sigma}}_{z\; 2}\left( {{\hat{a}}^{t} + \hat{a}} \right)}.}}}} & (1)\end{matrix}$

In this expression, ω_(r) and ω_(ai) are the frequencies of theoscillator and of the i^(th) qubit, respectively, while g₂(t) are thecorresponding longitudinal coupling strengths.

In the absence of external drives or of modulation of the coupling, thelongitudinal interaction only leads to a vanishingly small displacementof the oscillator field and can safely be dropped. This interaction canbe rendered resonant by modulating g_(i)(t) at the oscillator frequencyleading to a large qubit-state dependent displacement of the oscillatorstate. Measurement of the oscillator by homodyne detection can then beused for fast QND qubit readout. Consequently, modulating the couplingat the oscillator frequency rapidly dephases the qubits. To keepdephasing to a minimum, we instead use an off-resonant modulation ofg₂(t) at a frequency ω_(m) detuned from ω_(r) by many oscillatorlinewidths g₂(t)=g_(i) cos(ω_(m)t), where g_(1,2) are constant realamplitudes.

The oscillator-mediated qubit-qubit interaction can be made moreapparent by applying a polaron transformationÛ(t)=exp[Σ_(i=1,2)α_(i)(t){circumflex over (σ)}_(zi)â^(†)—H.c.] with anappropriate choice of α_(i)(t). Doing this, we find in the polaron framethe simple Hamiltonian

Ĥ _(pol)(t)=ω_(r) â ^(†) â+J _(z)(t){circumflex over(σ)}_(z1){circumflex over (σ)}_(z2).   (2)

The full expression for the {circumflex over (σ)}_(z){circumflex over(σ)}_(z)-coupling strength J_(z)(t) is given “Supplemental Material forFast and High-Fidelity Entangling Gate through Parametrically ModulatedLongitudinal Coupling” , which we will henceforth refer to as SM B. Inthe following we will, however, assume two conditions on the total gatetime, t_(g), such that this expression simplifies greatly. Forδt_(g)=n×2π and ω_(m)t_(g)=m×π, with n and in integers, we can replaceJ_(z)(t) by

$\begin{matrix}{{{\overset{\_}{J}}_{z} = {- {\frac{g_{1}g_{2}}{2}\left\lbrack {\frac{1}{\delta} + \frac{1}{\omega_{r} + \omega_{m}}} \right\rbrack}}},} & (3)\end{matrix}$

where δ≡ω_(r)−ω_(m) is the modulation drive detuning.

By modulating the coupling for a time t_(g)=θ/4/J_(z)|, evolution underEq. (2) followed by single qubit Z-rotations leads to the entanglingcontrolled-phase gate U_(CP)(θ) diag[1, 1, 1, e^(iθ)] Since U_(CP) (π)together with single qubit rotations forms a universal set, we onlyconsider this gate from now on.

Note that the conditions on the gate time used in Eq. (3) are notnecessary for the validity of Eq. (2), and the gate can be realizedwithout these assumptions. However, as we will discuss below, theseconditions are important for optimal gate performance: They ensure thatthe oscillator starts and ends in the vacuum state, which implies thatthe gate does not need to be performed adiabatically. Finally, notimposing the second constraint, ω_(m)t_(g)=m×π, only introduces fastrotating terms to Eq. (3) which we find to have negligible effect forthe parameters used later in this Letter. In other words, thisconstraint can be ignored under a rotating-wave approximation.

The above situation superficially looks similar to controlled-phasegates based on transversal coupling and strong oscillator driving. Thereare, however, several key differences. With transversal coupling, the{circumflex over (σ)}_(z){circumflex over (σ)}_(z) interaction isderived using perturbation theory and is thus only approximately validfor small g_(x)/{Δ, δ_(d)}, with Δ the qubit-oscillator detuning andδ_(d) the oscillator-drive detuning. For the same reason, it is alsoonly valid for small photon numbers n«n_(crit)=Δ²/4g² _(x). Moreover,this interaction is the result of a fourth order process ing_(x)/{Δ,δ_(d)}, leading to slow gates. Because of the breakdown of thedispersive approximation, any attempt to speed up the gate by decreasingthe detunings or increasing the drive amplitude results in low gatefidelities. In contrast, with longitudinal coupling, the {circumflexover (σ)}_(z){circumflex over (σ)}_(z) interaction is conveniently theresult of a second-order process in g_(1,2)/δ. In addition, since Eq.(2) is obtained after exact transformations on the system Hamiltonian,it is valid for any value of g_(1,2)/δ and is independent of theoscillator photon number. As will become clear later, this implies thatthe gate time and the gate infidelity can be decreased simultaneously.Finally, with longitudinal coupling, there is no constraint on the qubitfrequencies, in contrast with usual oscillator-induced phase gates wherethe detuning between qubits should preferably be small.

Oscillator-induced qubit dephasing.—FIG. 1 illustrates, for g₁=g₂, themechanism responsible for the qubit-qubit interaction. Underlongitudinal coupling, the oscillator field is displaced in aqubit-state dependent way, following the dashed lines in FIG. 1(a)(Panels (b) and (c) will be discussed later). This conditionaldisplacement leads to a non-trivial qubit

FIG. 1. (color online) Schematic illustration, in a frame rotating atω_(r), of the qubit-state dependent oscillator field in phase space forg₁=g₂ starting and ending in the vacuum state. The oscillator's pathsfor 100

and 111

are shown in dashed lines. The qubit-state dependent oscillator state isshown in light (t=t_(g)/4) and dark greys (t=t_(g)/2). The oscillator'sstate associated to {101

, 110

} stays in the vacuum state for the duration of the gate. (a) Nosqueezing. (b,c) Squeezing can help in erasing the which-qubit-stateinformation.

phase accumulation. This schematic illustration also emphasizes the maincause of gate infidelity for this type of controlled-phase gate,irrespective of its longitudinal or transversal nature: Photons leakingout from the oscillator during the gate carry information about thequbit state, leading to dephasing.

A quantitative understanding of the gate infidelity under photon losscan be obtained by deriving a master equation for the jointqubit-oscillator system. While general expressions are given in SM B. Tosimplify the discussion we assume here that g₁=g₂≡g. Following thestandard approach, the Lindblad master equation in the polaron framereads

$\begin{matrix}{{{\overset{.}{\rho}(t)} = {{- {i\left\lbrack {{\hat{H}}_{pol},{\rho (t)}} \right\rbrack}} + {{{\kappa }\left\lbrack \hat{a} \right\rbrack}{\rho (t)}} + {{\Gamma \left\lbrack {1 - {\cos \left( {\delta \; t} \right)}} \right\rbrack}{\left\lbrack {{\hat{\sigma}}_{z\; 1} + {\hat{\sigma}}_{z\; 2}} \right\rbrack}{\rho (t)}}}},} & (4)\end{matrix}$

where κ is photon decay rate and

[x] denotes the usual dissipation super-operator

[x]•=x•x^(†)−1/2{x^(†)x, •}. The last term of Eq. (4) corresponds to adephasing channel with rate Γ=2κ(g/2δ)². Since Ĥ_(pol) does not generatequbit-oscillator entanglement during the evolution, we can ensure that Γis the only source of dephasing by imposing that the initial and finalpolaron transformations do not lead to qubit-oscillator entanglement.This translates to the condition α_(i)(0)=α_(i)(t_(g))=0 and is realizedfor δt_(g)=n×2π, which is the constraint mentioned earlier (neglectingfast-rotating terms related to the second constraint ω_(m)t_(g)=m×π).More intuitively, it amounts to completing n full circles in FIG. 1, theoscillator ending back in its initial state. Note that these conclusionsremain unchanged if

FIG. 2. (color online) Average gate infidelity 1-

(full line) and gate time (dashed lines) of U_(CP)(π) as a function of(a) detuning, (b) coupling strength and (c) squeezing power. In panel(a) g/κ×10⁻³ is fixed at 2 (201), 3 (202), 4 (203). Note that thecorresponding three infidelity curves are indistinguishable on thisscale. In panel (b) δ/κ×10⁻⁵ is fixed at 0.75 (204), 1 (205), 1.25(206). In panel (c) parameters are δ/2π=0.6 GHz, g/2π=60 MHz, t_(g)=42.7ns, κ/2π=1 MHz. (207) rotating squeezing angle as illustrated in FIG.1(b). (208) squeezing at ω_(r) as illustrated in FIG. 1(c), andκ(ω_(m))=0 simulating a filter reducing the density of modes to zero atω_(m).

the oscillator is initially in a coherent state. As a result, there isno need for the oscillator to be empty at the start of the gate.

Based on the dephasing rate Γ and on the gate time t_(g), a simpleestimate for the scaling of the gate infidelity is 1-

˜Γ×t_(g)˜κ/δ[1]. A key observation is that this gate error isindependent of g, while the gate time scales as t_(g)˜δ/g². Both thegate time and the error can therefore, in principle, be made arbitrarilysmall simultaneously. This scaling of the gate error and gate time isconfirmed by the numerical simulations of FIG. 2, which shows thedependence of the gate infidelity on detuning δ and coupling strength g,as obtained from numerical integration of Eq. (4). The expected increasein both fidelity (full lines) and gate time (dashed lines) withincreasing detuning δ are apparent in panel (a). In addition, panel (b)confirms that, to a very good approximation, the fidelity is independentof g (full lines) while the gate time decreases as t_(g)˜1/g² (dashedlines).

This oscillator-induced phase gate can be realized in a wide range ofphysical platforms where longitudinal coupling is possible. Examplesinclude spin qubits in inhomogeneous magnetic field, singlet-tripletspin qubits, flux qubits capacitively coupled to a resonator andtransmon-based superconducting qubits. The parameters used in FIG. 2have been chosen following the latter references. In particular, takingκ/2π=0.01 MHz, g/2π=60 MHz and δ/2π=537 MHz results in a very short gatetime of t_(g)=37 ns with an average gate infidelity as small as 2×10⁻⁵.Taking into account finite qubit lifetimes T₁=30 μs and T₂=20 μs, wefind that the infidelity is increased to ˜10⁻³. In other words, the gatefidelity is limited by the qubit's natural decoherence channels withthese parameters.

A crucial feature of this gate is that the circular path followed by theoscillator field in phase space maximizes qubit-state dependent phaseaccumulation while minimizing de-phasing, allowing for high gatefidelities. Furthermore, we show below that this also allows forexponential improvement in gate fidelity with squeezing. It is thereforedesirable to minimize, or avoid completely, dispersive coupling inexperimental implementations [2].

Improved fidelity with squeezing—As discussed above, for fixed g and δthe fidelity increases with decreasing n. A small oscillator decay rateκ, however, comes at the price of longer measurement time if the sameoscillator is to be used for readout. This problem can be solved bysending squeezed radiation to the oscillator's readout port. Asschematically illustrated in FIG. 1, by orienting the squeezing axiswith the direction of the qubit-dependent displacement of the oscillatorstate, the which-path information carried by the photons leaving theoscillator can be erased. By carefully choosing the squeezing angle andfrequency, it is thus possible to improve the gate performance withoutreducing κ. We now show two different approaches to realize this,referring the reader to SM B for technical details.

A first approach is to send broadband two-mode squeezed vacuum at theinput of the oscillator, where the squeezing source is defined by a pumpfrequency ω_(p)=(ω_(r)+ω_(m))/2 and a squeezing spectrum with largedegree of squeezing at ω_(r) and ω_(m). A promising source of this typeof squeezing is the recently developed Josephson travelling waveamplifiers. With such a squeezed input field, a coherent state of theoscillator becomes a squeezed state with a squeezing angle that rotatesat a frequency δ/2. As illustrated in FIG. 1(b), this is precisely thesituation where the anti-squeezed quadrature and the displacement of theoscillator's state are aligned at all times. This leads to anexponential decrease in dephasing rate

Γ(r)˜e^(−2r)Γ(0),   (5)

with r the squeezing parameter. This reduction in dephasing rate leadsto the exponential improvement in gate fidelity with squeezing powershown by the brown line in FIG. 2(c). An interesting feature in thisFigure is that increasing κ by 2 orders of magnitude to allow for fastmeasurement, leads to the same ˜10⁻⁵ gate infidelity obtained abovewithout squeezing here using only ˜6 dB of squeezing. Since numericalsimulations are intractable for large amount of squeezing, we depict theinfidelity obtained from a master equation simulation by a solid lineand the expected infidelity from analytical calculations by adash-dotted line.

An alternative solution is to use broadband squeezing centered at theoscillator's frequency, i.e. a squeezing source defined by a pumpfrequency ω_(p)=ω_(r). As illustrated in FIG. 1(c), using this type ofinput leads to a squeezing angle that is constant in time in a framerotating at ω_(r). With this choice, information about the qubits' statecontained in the â^(†)+â quadrature of the field is erased whileinformation in the i(â^(†)−â) quadrature is amplified (cf. FIG. 1). Byitself, this does not lead to a substantial fidelity improvement.However, a careful treatment of the master equation shows that Eq. (5)can be recovered by adding a filter reducing the density of modes atω_(m) to zero at the output port of the oscillator. Filters of this typeare routinely used experimentally to reduce Purcell decay ofsuperconducting qubits. As illustrated by the dark blue line in FIG.2(c), using single-mode squeezing at ω_(r) and a filter at themodulation frequency, we recover the same exponential improvement foundwith two-mode squeezing, Eq. (5), in addition to a factor of twodecrease in gate infidelity without squeezing.

Interestingly, rotating the squeezing axis by π/2 when squeezing at theoscillator frequency helps in distinguishing the different oscillatorstates and has been shown to lead to an exponential increase in thesignal-to-noise ratio for qubit readout. In practice, the differencebetween performing a two-qubit gate and a measurement is thus theparametric modulation frequency (off-resonant for the gate and onresonance for measurement) and the choice of squeezing axis.

We note that Eq. (5) was derived from a master equation treatment underthe standard secular approximation, which is not valid at high squeezingpowers (here, ≳14 dB). At such high powers, the frequency dependence ofn together with other imperfections are likely to be relevant.

Circuit QED implementation.—The proposed gate is applicable in a varietyof physical systems where Eq. (1) is realizable. As examples, we mentionflux qubits capacitively coupled to a resonator and spin qubits . Wehere propose an alternative realization based on transmon qubits, whichare particularly attractive due to their excellent coherence times. Thecircuit we consider is illustrated in FIG. 3, and more details are givenin SM B. This realization does not, in fact, give an exact realizationof Eq. (1), as we show below that there are additional terms in theeffective Hamiltonian, but these terms turn out to not be of anydetrimental consequence for the gate. This circuit QED realization withtransmon qubits is an extension to two qubits of the circuit introducedin Paper A. A detailed derivation of the Hamiltonian of this circuityields in the two-level approximation

$\begin{matrix}{{{{\hat{H}}^{circ}(t)} = {{\omega_{r}{\hat{a}}^{\dagger}\hat{a}} - {{ɛ(t)}\left( {\hat{a} + {\hat{a}}^{\dagger}} \right)} + {\sum\limits_{i}\left\lbrack {{\frac{\omega_{ai}}{2}{\hat{\sigma}}_{zi}} + {{g_{i}(t)}\left( {\hat{a} + {\hat{a}}^{\dagger}} \right){\hat{\sigma}}_{zi}} + {\chi_{i}{\hat{a}}^{\dagger}\hat{a}{\hat{\sigma}}_{zi}}} \right\rbrack}}},} & (6)\end{matrix}$

where modulation of the external fluxes indicated in FIG. 3 leads to thedesired modulation of the longitudinal coupling, g_(i)(t)˜Φ_(i)(t), inaddition to a small renormalization of the qubits frequencies. A largedrive on the resonator leads to a similar effect, which means that inpractice the longitudinal couplings can be modulated either throughexternal fluxes or a strong resonator drive.

This Hamiltonian is different from Eq. (1) due to the presence of theλ_(i) ac-Stark shifts. We have also included a drive term ε(t), whichincludes a flux-induced contribution, but can be tuned via a gatevoltage applied to the resonator (not shown in the figure). Note thatthe χ_(i)-terms are not the result of a dispersive approximation to aJaynes-Cummings Hamiltonian but appear naturally in this circuit anddoes not introduce any Purcell decay or unwanted resonator-qubithybridization. For reasonable circuit parameters, we find thatχ_(i)˜g_(i), which means that these ac-Stark shifts cannot be neglected.However, a {circumflex over (σ)}_(z1){circumflex over (σ)}_(z2)interaction can be made explicit through a polaron-type transformationwhich leads, in the symmetric case χ₁=χ2≡χ, g₁=g₂≡g (taken forsimplicity), to the transformed

Hamiltonian

$\begin{matrix}{{{{\hat{H}}_{pol}^{circ}(t)} = {{\delta {\hat{a}}^{\dagger}\hat{a}} + {\chi {\hat{a}}^{\dagger}{\hat{a}\left( {\sigma_{z\; 1} + \sigma_{z\; 2}} \right)}} + {{J_{z}(t)}\sigma_{z\; 1}\sigma_{z\; 2}} + {\frac{\varpi (t)}{2}\left( {{\hat{\sigma}}_{z\; 1} + {\hat{\sigma}}_{z\; 2}} \right)}}},} & (7)\end{matrix}$

where full expressions for J_(z)(t) and ω=(t) can be found in SM B. Thisequation is very similar to Eq. (2) in the sense that the ac-Stark shiftdoes not play a role since the resonator stays in its ground state inthis frame (assuming an initial vacuum state). The additional singlequbit Z rotations can be corrected for through standard single-qubitgates.

There are however two main differences with Eq. (2). First, we have toimpose an additional condition 2χt_(g)=2π×p with p an integer to avoidresidual resonator-qubit entanglement at the end of the gate. Then,J_(z)(t) can be replaced by

$\begin{matrix}{{J_{z\; 0} = \frac{- \left( {{g\; \delta} + {ɛ\chi}} \right)^{2}}{2{\delta \left( {\delta^{2} - {4\chi^{2}}} \right)}}},} & (8)\end{matrix}$

where we have assumed the ε(t)=ε cos(ω_(m)t) real, for simplicity. Thesecond difference is that the paths in phase space differ from the pathsshown in FIG. 1. Unfortunately we find that in the presence of ac-Starkshift, squeezing cannot be used to reduce the dephasing ratesubstantially since it is not possible to align the anti-squeezedquadrature and the displacement of the field at all times. Thesedifferences aside, the gate time and infidelity stay approximately thesame in the absence of squeezing. For example, taking κ/2π=0.01 MHz,g/2π=40 MHz, χ/2π=75 MHz, ε=158 MHz and δ/2π=600 MHz results in a gatetime of t_(g)=40 ns with an average gate infidelity of 5×10⁻⁵, which iscomparable to the numbers based on Eq. (1).

Independent control of the couplings can be used for individual qubitreadout while single-qubit X and Y rotations can be realized by drivingweakly coupled voltage gates (not shown in the figure). It isinteresting to note that longitudinal coupling can alternatively beobtained from the dispersive regime of transveral coupling underappropriate resonator drive. This approximate realization oflongitudinal coupling however suffers from all of the above mentioneddrawbacks of the transversal coupling from which it emerges.

Scalability.—So far we have focused on two qubits coupled to a singlecommon oscillator. Longitudinal coupling of several qubits to separateoscillators that are themselves coupled transversely has favorablescaling properties. Interestingly, the gate introduced in this Letter

FIG. 3. (Color online.) Circuit QED implementation of longitudinalinteraction with two qubits coupled to a common LC oscillator. Furtherdetails on the mapping of this circuit to Eq. (6) can be found in SM B.

can also be implemented in such an architecture. Consider two qubitsinteracting with distinct, but coupled, oscillators with thecorresponding Hamiltonian

$\begin{matrix}{{\hat{H}}_{ab} = {{\omega_{a}{\hat{a}}^{\dagger}\hat{a}} + {\omega_{b}{\hat{b}}^{\dagger}\hat{b}} + {\frac{1}{2}\omega_{a\; 1}{\hat{\sigma}}_{z\; 1}} + {\frac{1}{2}\omega_{a\; 2}{\hat{\sigma}}_{z\; 2}} + {{g_{1}(t)}{{\hat{\sigma}}_{z\; 1}\left( {{\hat{a}}^{\dagger} + \hat{a}} \right)}} + {{g_{2}(t)}{{\hat{\sigma}}_{z\; 2}\left( {{\hat{b}}^{\dagger} + \hat{b}} \right)}} - {{g_{ab}\left( {{\hat{a}}^{\dagger} - \hat{a}} \right)}{\left( {{\hat{b}}^{\dagger} - \hat{b}} \right).}}}} & (9)\end{matrix}$

In this expression, â, {circumflex over (b)} label the mode of eachoscillator of respective frequencies ω_(a,b), and g_(ab) is theoscillator-oscillator coupling. As above, g_(1,2)(t) are modulated atthe same frequency ω_(m), corresponding to the detuningsδ_(a)≡ω_(a)−ω_(m) and δ_(b)≡ω_(b)−ω_(m). Following the same procedure asabove and performing a rotating-wave approximation for simplicity, wefind a Hamiltonian in the polaron frame of the same form as Eq. (2), butnow with a modified {circumflex over (σ)}_(z){circumflex over (σ)}_(z)interaction strength

$\begin{matrix}{{{\overset{\_}{J}}_{z} = {\frac{1}{2}\frac{g_{1}g_{2}g_{ab}}{{\overset{\_}{\delta}}^{2} - {g_{ab}^{2}\left( {1 + \zeta^{2}} \right)}}}},} & (10)\end{matrix}$

where δ=(δ_(a)+δ_(b))/2 and ζ=(ω_(b)−ω_(a))/(2g_(ab)) Thisimplementation allows for a modular architecture, where each unit cellis composed of a qubit and coupling oscillators, used for both readoutand entangling gates. Such a modular approach can relax designconstraints and avoids spurious interactions with minimal circuitcomplexity.

Conclusion.—We have proposed a controlled-phase gate based on purelylongitudinal coupling of two qubits to a common oscillator mode. The keyto activating the qubit-qubit interaction is a parametric modulation ofthe qubit-oscillator coupling at a frequency far detuned from theoscillator. The gate infidelity and gate time can in principle be madearbitrarily small simultaneously, in stark contrast to the situationwith transversal coupling. We have also shown how the gate fidelity canbe exponentially increased using squeezing and that it is independent ofqubit frequencies. The gate can moreover be performed remotely in amodular architecture based on qubits coupled to separate oscillators.Together with the fast, QND and high-fidelity measurement schemepresented in Paper A, this makes a platform based on parametricmodulation of longitudinal coupling a promising path towards universalquantum computing in a wide variety of physical realizations.

[1] Note that 1-

refers only to the error due to photon decay, excluding the qubitsnatural T₁ and T₂ times.

[2] This can be done by reducing the participation ratio, η, such thatx≤κ.

1. A circuit quantum electrodynamics (circuit QED) implementation of acontrol-phase quantum logic gate U_(CP)(θ)=diag[1,1,1,e^(iθ)], thecircuit QED implementation comprising: two qubits Q_(i), where i=1corresponds to a first qubit Q₁ and i=2 corresponds to a second qubitQ₂, each having a frequency ω_(qi) and being characterized by{circumflex over (σ)}_(zi); a first resonator R_(a), associated with thequbit Q₁, defined by: a resonator frequency ω_(ra); a resonatorelectromagnetic field characterized by â^(†) and â; a longitudinalcoupling strength g_(1z) with the qubit Q₁; a first longitudinalcoupling g_(1z){circumflex over (σ)}_(1z)(â^(†)+â); wherein a secondresonator R_(b), such that, when the second resonator R_(b) isindependent from R_(a): R_(b) is associated with the qubit Q₂; alongitudinal resonator-resonator coupling g_(ab) is defined and R_(b) isfurther defined by: a second resonator frequency ω_(rb); a secondresonator electromagnetic field characterized by {circumflex over(b)}^(†) and {circumflex over (b)}; a second longitudinal couplingstrength g_(2z) with the qubit Q₂; a second longitudinal couplingg_(2z){circumflex over (σ)}_(2z)({circumflex over (b)}^(†)+{circumflexover (b)}); and when the second optional resonator R_(b) is notindependent from R_(a) and integrated into R_(a): R_(a) is associatedwith the qubit Q₂; the longitudinal resonator-resonator couplingg_(ab)=1; the second resonator electromagnetic field is characterized byâ^(†) and â where {circumflex over (b)}^(†)=â^(†) and {circumflex over(b)}=â; the second resonator frequency ω_(rb)=ω_(ra); the secondlongitudinal coupling strength g_(2z) is between the qubit Q₂ and R_(a);the second first longitudinal coupling g_(2z){circumflex over(σ)}_(2z)({circumflex over (b)}^(†)+{circumflex over (b)}) is defined byR_(a) as g_(2z){circumflex over (σ)}_(2z)(â^(†)+â); and a modulatorperiodically modulating, at a frequency ω_(m) during a time t, thelongitudinal coupling strengths g_(1z) and g_(2z) with respectivesignals of respective amplitudes {tilde over (g)}₁ and {tilde over(g)}₂, wherein selecting a defined value for each of t, g_(1z) andg_(2z) determines θ to specify the quantum logical operation performedby the control-phase quantum logic gate and wherein when the qubit Q₁and the qubit Q₂ are decoupled when either one of the defined value ofg_(1z) and the defined value of g_(2z) is to set to
 0. 2. The circuitQED implementation of claim 1, further comprising a transmitter forselectively providing a modulator activation signal to the modulator foractivating the modulator for the duration t.
 3. The circuit QEDimplementation of claim 1, further comprising a signal injectorproviding a squeezed input to diminish a which-qubit-state information.4. The circuit QED implementation of claim 3, wherein the squeezed inputis a single-mode squeezed input.
 5. The circuit QED implementation ofclaim 3, wherein the squeezed input is a two-mode squeezed input.
 6. Thecircuit QED implementation of claim 3, wherein the signal injectorrelies on broadband squeezed centered at ω_(rb) and/or ω_(ra).
 7. Thecircuit QED implementation of claim 1, wherein the qubit Q₁ and thequbit Q₂ are transmons each comprising two Josephson junctions withrespectively substantially equivalent capacitive values and themodulator comprises an inductor-capacitor (LC) oscillator, thelongitudinal coupling resulting from mutual inductance between theoscillator and the transmons, the oscillator varying a flux φ₁ in thequbit Q₁ and a flux φ₂ in the qubit Q₂.
 8. The circuit QEDimplementation of claim 7, wherein a 3-Wave mixing Josephson dipoleelement is used to couple the qubit Q₁ and the resonator R_(a).
 9. Amethod for specifying a quantum logical operation performed by acontrol-phase quantum logic gate U_(CP)(θ) =diag[1,1,1,e^(iθ)], whereinthe circuit QED implementation comprises (I) two qubit Q₁, where i=1corresponds to a first qubit Q₁ and i=2 corresponds to a second qubitQ₂, each having a frequency ω_(qi) and being characterized by{circumflex over (σ)}_(zi); (II) a first resonator R_(a), associatedwith the qubit Q₁, defined by a first resonator frequency ω_(ra), afirst resonator electromagnetic field characterized by â^(†) and â, afirst longitudinal coupling strength g_(1z) with the qubit Q₁ and afirst longitudinal coupling g_(1z){circumflex over (σ)}_(1z)(â^(†)+â);(III) a second resonator R_(b), such that, when the second resonatorR_(b) is independent from R_(a), R_(b) is associated with the qubit Q₂,a longitudinal resonator-resonator coupling g_(ab) is defined and R_(b)is further defined by: a second resonator frequency 107 _(rb), a secondresonator electromagnetic field characterized by {circumflex over(b)}^(†) and {circumflex over (b)}, a second longitudinal couplingstrength g_(2z) with the qubit Q₂, a second longitudinal couplingg_(2z){circumflex over (σ)}_(2z)({circumflex over (b)}^(†)+{circumflexover (b)}) and (IV), when the second optional resonator R_(b) is notindependent from R_(a) and integrated into R_(a), R_(a) is associatedwith the qubit Q₂, the longitudinal resonator-resonator couplingg_(ab)=1, the second resonator electromagnetic field is characterized byâ^(†) and â where {circumflex over (b)}^(†)=â^(†) and {circumflex over(b)}=â, the second resonator frequency ω_(rb)=ω_(ra), the secondlongitudinal coupling strength g_(2z) is between the qubit Q₂ and R_(a),the second longitudinal coupling g_(2z){circumflex over(σ)}_(2z)({circumflex over (b)}^(†)+{circumflex over (b)}) is defined byR_(a) as g_(2z){circumflex over (σ)}_(2z)(â^(†)+â), the methodcomprising: periodically modulating, at a frequency ω_(m) during a timet, the longitudinal coupling strengths g_(1z) and g_(2z) with respectivesignals of respective amplitudes {tilde over (g)}₁ and {tilde over(g)}₂; selecting a defined value for each of t, g_(1z) and g_(2z)thereby fixing θ to specify the quantum logical operation performed bythe control-phase quantum logic gate; and setting at least one of thedefined value of g_(1z) and the defined value of g_(2z) is to 0 todecouple the qubit Q₁ from the qubit Q₂.
 10. The method of claim 9,wherein selecting the defined value for each of t, g_(1z) and g_(2z)comprise a selectively providing a modulator activation signal to themodulator for activating the modulator for the duration t.
 11. Themethod of claim 9 or claim 10, further comprising providing a squeezedinput to diminish a which-qubit-state information.
 12. The method ofclaim 11, wherein the squeezed input is a single-mode squeezed input.13. The method of claim 11, wherein the squeezed input is a two-modesqueezed input.
 14. The method of claim 11, wherein the squeezed inputrelies on broadband squeezed centered at ω_(rb) and/or ω_(ra).
 15. Themethod of claim 9, wherein the qubit Q₁ and the qubit Q₂ are transmonseach comprising two Josephson junctions with respectively substantiallyequivalent capacitive values and the modulator comprises aninductor-capacitor (LC) oscillator, the longitudinal coupling resultingfrom mutual inductance between the oscillator and the transmons, theoscillator varying a flux φ₁ in the qubit Q₁ and a flux φ₂ in the qubitQ₂.
 16. The method of claim 15, wherein a 3-Wave mixing Josephson dipoleelement is used to couple the qubit Q₁ and the resonator R_(a).